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Compact Error-Resilient Computational DNA Tiling Assemblies

John H.Reif, Sudheer Sahu, and Peng Yin

Presenter: Seok, Ho-SIK

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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

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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

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*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)

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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

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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 scal

e assemblies is to construct error-resilient tiles

Winfree’s approach resulted in a final structure that is f

our times the size of the original oneWinfree wanted to inhibit occurrence of an incorrectly matched tile structure

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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 positi

on (i+1,j) U(i,j)= U(i-1,j) OP2 V(i,j-1)

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)

U(i-1,j)

V(i,j)U(i,j)

V(i,j-1)

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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

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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 padU(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!

Error checking portion

V(i,j), V(i-1,j)

U(i-2,j),

V(i-1,j)

V(i,j-1), V(i-1,j-1)

U(i-1,j), V(i,j)

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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 seq

uences 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 struct

ure We are talking about some nice kind of nano machine not damn it DN

A machine!!!

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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)

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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 portionInformation

Information

Some kind of linker of fastener

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Error analysis – 3 (Proof)1. 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)

Error checking portion

V(i,j), V(i-1,j)

U(i-2,j),

V(i-1,j)

V(i,j-1), V(i-1,j-1)

In this case, T1(i, j) will compute incorrect value for right V(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)1.2 Error is due to the incorrect value of the le

ft 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)

U(i-1, j)V(i, j)

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

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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)

Error checking portion

V(i,j), V(i-1,j)

U(i-2,j),

V(i-1,j)

V(i,j-1), V(i-1,j-1)

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-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

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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 right is 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)=62, 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

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*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

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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

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Comparison to Winfree Winfree’s method can inhibit the spreading of incorrect m

ismatches With their notion of dependency, they (Reif & his followers) do not

show the method for spreading inhibitionTheir 62 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