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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
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
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Assembly with no error corrections
U(i,j)

NXM array

V(i,j)
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)

V(i,j-1)
U(i-1,j)

U(i,j)
Boolean value communicated to the
position (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)
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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


V(i,j), V(i-1,j)
U(i-2,j),
U(i-1,j),
V(i,j)
Note
V(i-1,j)


Error
checking
portion
V(i,j-1), V(i-1,j-1)
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!


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
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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
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!!!
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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 portion
Information
Some kind of linker
of fastener
Information
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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)
V(i,j), V(i-1,j)
U(i-2,j),
U(i-1, j)
V(i, j)
V(i-1,j)
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
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)
Error
checking
portion
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
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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)
V(i,j), V(i-1,j)
In this case, T1(i, j) will compute incorrect value
U(i-2,j),
V(i-1,j)
U(i-1, j)
V(i, j)
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))
Error
checking
portion
V(i,j-1), V(i-1,j-1)
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)=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
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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
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
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