Assembling Systems Jacob Hendricks University of Wisconsin River - - PowerPoint PPT Presentation

Introduction to Modeling Algorithmic Self- Assembling Systems

Dagstuhl Seminar: Algorithmic Foundations of Programmable Matter

Jacob Hendricks

University of Wisconsin – River Falls

Self-assembly

• Self-assembly: the process by which relatively

simple components in a disorganized state autonomously combine to form more complex

• bjects
• Natural self-assembling systems span the spectrum of

complexity

• From crystals (passive components with simpler geometries)
• To biology (active components with more complex geometries,

properties, and behaviors)

• We try to mimic these using artificially

developed/modeled systems

• Algorithmic self-assembly
• Based on computational theory
• Design of components forces the self-assembly process to follow

prescribed algorithms

Self-assembly in Nature

Virus from: https://micro. magnet.fsu.edu/cells/virus.html From: https://cheerioseffect.ca/ Photographer: Richard Ling

DNA Self-assembly

Tile Assembly Model

A highly abstracted representation of DNA motifs which can be treated logically as 2-dimensional squares

Molecular structure of a Holliday junction (Image courtesy of Wikipedia) With the ‘sticky ends’ treated as glues, these molecules can be thought of as square ‘tiles’ Schematic view with extended ‘sticky ends’

Ned Seeman Erik Winfree

Abstract Tile Assembly Model (aTAM)

• Fundamental components are 2-D

square tiles

• Each side has an associated glue,

with:

• A type (usually represented by a

string value)

• An integer-valued strength

(usually 0, 1, or 2)

• Tiles can also have labels (non-

functional, for convenience)

• Tiles cannot be rotated
• Finite number of different tile types
• An infinite supply of each tile type
• Abutting sides of tiles bind if both

glue strengths and values match

• Those sides bind with that shared

strength

• A tile can bind to an assembly if

the sum of binding strengths is at least equal to the “temperature” value of the system (usually 1 or 2)

• Assembly begins from a “seed” tile or

assembly and grows 1 tile at a time

Tile Assembly System in the aTAM

Temperature value = 2 Seed = (S, (0,0)) Tile set:

S

Tile Assembly System in the aTAM

Temperature value = 2 Seed = (S, (0,0)) Tile set:

S

Producible Assembly:

S

Tile Assembly System in the aTAM

Temperature value = 2 Seed = (S, (0,0)) Tile set:

S

Producible Assembly:

S

Terminal Assembly This system is directed.

Tile Assembly System in the aTAM

Temperature value = 2 Seed = (S, (0,0)) Tile set:

Tile Assembly System in the aTAM

Tile set: Temperature value = 2 Seed = (S, (0,0))

Attachment by 2 strength-1 bonds is a form of “cooperation” between multiple tiles that gives the model great power

Tile Assembly System in the aTAM

Tile set: Temperature value = 2 Seed = (S, (0,0))

Tile Assembly System in the aTAM

Tile set: Temperature value = 2 Seed = (S, (0,0))

Tile Assembly System in the aTAM

Tile set: Temperature value = 2 Seed = (S, (0,0))

Tile Assembly System in the aTAM

Tile set: Temperature value = 2 Seed = (S, (0,0))

Θ log 𝑂 tile types Θ 𝑂 Θ log 𝑂

DNA Binary Counters

[Evans, 2014]

Tile Assembly System in the aTAM

Tile set:

Temperature value = 2 Seed = (S, (0,0))

Tile Assembly System in the aTAM

Tile set:

Temperature value = 2 Seed = (S, (0,0))

Tile Assembly System in the aTAM

Tile set:

DNA Sierpinski Triangle

[Papadakis, Rothemund, Winfree 2004]

scale bars = 100 nm

Sierpinksi’s Triangle

Strict vs. Weak Self-Assembly

There are two main notions of the self-assembly of a set of points (defining a shape or pattern) Strict self-assembly Every point in the set, and no other point, receives a tile. (Note that this means that the shape must be connected.) Weak self-assembly Every point in the set, and no other point, receives a tile of a type from a specially defined subset of tile types (e.g. `red’ tile types). Other points may receive tiles of the other tile types.

Strict vs. Weak Self-Assembly

There are two main notions of the self-assembly of a set of points (defining a shape or pattern) Strict self-assembly Every point in the set, and no other point, receives a tile. (Note that this means that the shape must be connected.) Weak self-assembly Every point in the set, and no other point, receives a tile of a type from a specially defined subset of tile types (e.g. `red/blue’ tile types). Other points may receive tiles of the

• ther tile types.

Sierpinksi’s Triangle

Sierpinski Triangle:

• Easy to weakly self-

assemble: only 7 tile types

• Impossible to strictly

self-assemble [Lathrop, Lutz, Summers, TCS 2009]

• Approximation is

possible [Patitz, Summers, DNA 14]

• “Similar” shape
• Same fractal

dimension (1.58)

Fibered Sierpinski Triangle

[Patitz and Summers 2010]

Algorithmic Self-assembly

Fujibayashi, Hariadi, Park, Winfree, & Murata, ACS NanoLetters, 8-7 (2007)

Self-similar fractal

Barish, Shulman, Rothemund, and Winfree, PNAS, 106 (2009)

Binary counter

• Goals of algorithmic self-assembly include:
• Embedding computations which control the

dimensions, shapes, and compositions of assemblies

• Discovering powers and limitations of self-

assembling systems

• Information theoretic lower bounds (e.g. an n x n square

requires log(n)/log(log(n)) unique tile types) [Rothemond, Winfree, STOC 2000]

• Impossibility of self-assembling some structures (e.g strict

version of Sierpinski triangle [Lathrop, Lutz, Summers, CiE 2007]

• Designing systems which are robust to errors
• Model a variety of different phenomena

Fibered Sierpinski Carpet

[Patitz and Summers, Natural Computing 2010]

Tile Assembly Model

• Design concerns for a tile assembly system (TAS)
• Tile complexity: How many unique tile types are

required?

• Scale factor: Does it form a scaled version of the

desired shape? If so, what is the scaling factor?

• It is directed – meaning that despite nondeterministic

assembly sequences, it always produces the same thing

2 1 3 4 2 1 3 4 X 5 6 7 8 9 tile types 6 tile types! X X X X Scale factor = 2

S

Self-assembly Models

• Variations on models
• Differing dynamics
• Hierarchical self-assembly
• Staged self-assembly
• Varying temperatures and concentrations
• Allow tile rotation, reflection, etc.
• Cooperative vs. non-cooperative
• Allowing errors
• Differing components
• Polyominoes and polygons
• Repulsive “glues”
• Active, signal-passing tiles

[Cannon et al. 2013] [Staged Self-Assembly, Demain et al. 2008] [Kao, Schweller 2006; Summers 2009 (figure)] [Kao, Schweller 2008; Doty 2010 (figure)] [Hendricks, Patitz, Rogers DNA 21] [Cook et al. SODA 2011] [Meunier UCNC 2014] [Doty et al. FOCS 2010] [Patitz et al. DNA 17] [Demaine et al. ICALP 2014] [SODA 2015] [Fekete et al, SODA 2014] [Padilla et al. UCNC 2013]

Benchmarks of a self-assembly model

• Universal computation
• Algorithmic behavior is possible
• aTAM is universal [Winfree 1998]
• Efficient assembly of NxN square
• Algorithmic behavior is useful
• Fewer tile types = more efficient

Benchmarks of a self-assembly model

• Universal computation
• Algorithmic behavior is possible
• aTAM is universal [Winfree 1998]
• Efficient assembly of NxN square
• Algorithmic behavior is useful
• Fewer tile types = more efficient

Turing Machines Simulation by a Zig-Zag Tile Assembly System

Temperature 2

Turing Machines Simulation by a Zig-Zag Tile Assembly System

Temperature 2

Turing Machines Simulation by a Zig-Zag Tile Assembly System

Temperature 2

Turing Machines Simulation by a Zig-Zag Tile Assembly System

Turing Machines Simulation by a Zig-Zag Tile Assembly System

Turing Machines Simulation by a Zig-Zag Tile Assembly System

Benchmarks of a self-assembly model

• Universal computation
• Algorithmic behavior is possible
• aTAM is universal [Winfree 1998]
• Efficient assembly of NxN square
• Algorithmic behavior is useful
• Fewer tile types = more efficient

Efficient assembly of NxN square

N N

Temperature 1

Efficient assembly of NxN square

N N

Temperature 1

2N – 1 Tile Types

Efficient assembly of NxN square

Temperature 2

Θ 𝑂 Θ 𝑂

Tile types:

Θ log 𝑂 Ω log 𝑂 log log 𝑂

[Rothemund, Winfree 2000] [Adleman, Cheng, Goel, Huang, STOC 2001]

Non-Cooperative Self-Assembly

• Only one glue needs to match for a tile to attach
• Example: Temperature-1 aTAM

Tile Assembly System in the aTAM

Temperature value = 1 Seed = (S, (0,0)) Tile set:

S

Producible Assembly:

Non-Cooperative Self-Assembly

• Conjectured to be “weak” in the general TAM

[Doty, Patitz, Summers, Theoretical Computer Science, 2011]

• Computation impossible
• Cannot be algorithmically directed
• Huge numbers of (sometimes exponentially more) unique tile

types required to self-assemble basic shapes

• Pierre-Etienne Meunier showed that tile sets of n tile types

can create assemblies of diameter 2n+o(n) [UCNC 2015]

Non-Cooperative Self-Assembly

• Computation is conjectured to be impossible for temperature-1

aTAM

• Theoretical results have shown a large number of workarounds:
• 3-D [Cook, Fu, Schweller, SODA 2011]
• Probabilistic [Cook, Fu, Schweller, SODA 2011]
• Negative strength glues [Patitz, Schweller, Summers, DNA 17]
• Polyominoes [Fekete, Hendricks, Patitz, Rogers, Schweller,

SODA 2015]

• Regular Polygons [Hendricks, Patitz, Rogers, SODA 2016]

Bit Reading Gadgets

Assemblies that encode bit values (0 or 1)

1

Bit Reading Gadgets

Paths grow past the dark gray subassembly and “read” the encoded bit values

1

Bit Reading Gadgets

Bit Reading Gadgets

First, we show that a polyTAM system is capable of writing and

reading bits at temperature 1.

Here’s the idea with a polyomino consisting of 5 squares.

Bit Reading Gadgets

1 is represented as: 0 is represented as:

How do we “read” these bits?

Bit Reading Gadgets

Reading a 1 at temperature 1.

1

1 has been read.

Bit Reading Gadgets

Reading a 0 at temperature 1.

1

0 has been read.

Computing with Bit Reading Gadgets

Self-assembly Models

• Variations on models
• Differing dynamics
• Hierarchical self-assembly
• Staged self-assembly
• Varying temperatures and concentrations
• Allow tile rotation, reflection, bending, etc.
• Cooperative vs. non-cooperative
• Allowing errors
• Differing components
• Polyominoes and polygons
• Repulsive “glues”
• Active, signal-passing tiles

[Cannon et al. 2013] [Staged Self-Assembly, Demain et al. 2008] [Kao, Schweller 2006; Summers 2009 (figure)] [Kao, Schweller 2008; Doty 2010 (figure)] [Cook et al. SODA 2011] [Meunier UCNC 2014] [Doty et al. FOCS 2010] [SODA 2015] [Fekete et al, SODA 2014] [Patitz et al. DNA 17] [Padilla et al. UCNC 2013]

Comparing Models with Simulation

• Can systems in one model simulate systems in

another model?

• Can a single tile set in one model be used to simulate

any system in another model?

2-Handed Tile Assembly Model (2HAM)

Temperature 2

What do these tiles do?

System Simulation

Each tile of the original system Each tile of the original system may be simulated by one or more blocks of tiles in the simulator

Simulated system: T Simulating system (simulator): S

System Simulation

Simulated system: T Simulating system (simulator): S

If and only if it can assemble in the original system T, it can assemble in the simulator S

System Simulation

Simulated system: T Simulating system (simulator): S

If and only if it can assemble in the original system T, it can assemble in the simulator S

2HAM vs aTAM

The 2HAM can simulate the aTAM

• For every aTAM system, there exists a 2HAM system which

can simulate with only a constant increase in scale factor

• Forces the small macrotiles of the 2HAM simulator to

follow aTAM dynamics, i.e. only attach one at a time to an assembly containing the seed

[Cannon, Demaine, Demaine, Eisenstat, Patitz, Schweller, Summers, Winslow, STACS 2013]

Intrinsic Universality

• Intrinsically universal

(IU) model:

• A system within the

model can simulate all

• ther systems
• May require scaling
• A function exists to map

the simulating system to the simulated system

• Must simulate full

behavior, not just achieve same end result

• IU models:
• Turing machines
• Cellular automata
• Abstract Tile Assembly

Model [Doty, Lutz, Patitz, Schweller, Summers, and Woods, FOCS 2012]

• Not IU
• 2HAM [Demaine et al.

ICALP 2013]

• The class of directed

aTAM systems [Hendricks, Patitz, Rogers, FOCS accepted]

The aTAM is Intrinsically Universal

We define a tile set U which can be used to create an aTAM system which simulates an abritrary aTAM system T T

• U operates at temperature 2, but can simulate any temperature
• U can simulate every possible assembly sequence of T

T =(T,σ,t) is a TAS UT = (U, σT, 2) simulates T by forming ‘supertiles’ (square assemblies consisting of tiles from U) which correspond to the tiles in T

Self-assembly Models

[Figure from Intrinsic universality and the computational power of self-assembly. Woods 2013]

Signal Tile Assembly Model (STAM)

• Extension of the 2HAM
• Tiles are allowed to have sets of glues on each edge
• Tiles have an initial state where each glue is either on
• r latent
• With each glue we associate a set of actions (i.e.

turning other glues on the tile on or off) that are performed asynchronously upon binding

• Each signal can only “fire” once

[Padilla, Patitz, Pena, Schweller, Seeman, Sheline, Summers, Zhong. UCNC 2013]

Signal Tile Assembly Model (STAM)

What do these tiles do?

Signal Tile Assembly Model (STAM)

“A Signal-Passing DNA-Strand-Exchange Mechanism for Active Self-Assembly of DNA Nanostructures”, Padilla, Jennifer E. and Sha, Ruojie and Kristiansen, Martin and Chen, Junghuei and Jonoska, Natasha and Seeman, Nadrian C., Angewandte Chemie International Edition, 1521-3773 (2015)

Signal Tile Assembly Model (STAM)

• Can “efficiently” simulate Turing machines
• i.e. each simulated step of the Turing machine requires only a

constant number of additional tiles

• Can strictly self-assemble the Sierpinski triangle
• Can force tile complexity into “signal complexity”
• i.e. constant sized tile sets can make different shapes by

increasing the number of signals per tile

• For each temperature, there exists an STAM tile set which

has no more than 1 signal per tile and which is IU for the class of all STAM systems at that temperature which don’t deactivate glues

• These IU tile sets can be simulated by regular (non-signal) 3D

tiles [Hendricks, Padilla, Patitz, Rogers, DNA 19]

Signal Tile Assembly Model (STAM)

Many fractals strictly self-assemble in the STAM at temperature 1, including Sierpinski’s triangle. [DNA 22 (Munich)]

The 22nd International Conference on DNA Computing and Molecular Programming will be held September 4-8, 2016 in Munich, Germany, at Ludwig-Maximilians-Universität (LMU).