Self-Assembly of Spherical Particles Natalie Arkus, Vinothan N. Manoharan, Michael Brenner School of Engineering and Applied Sciences Harvard University

Self-Assembly of Spherical Particles

Natalie Arkus, Vinothan N. Manoharan, Michael Brenner

School of Engineering and Applied Sciences

Harvard University

HIV viral shellCircuit

Assembly Mechanisms

Man-made ‘Natural’

Spontaneous Step by step

Can we manipulate natural assembly to make ‘man-made’ objects?

A Model System for Self-Assembly

Guangnan Meng

•Identical spheres

•Don’t overlap

•Stick to one another

Sphere diameter ≈ 1 micron

Spheres = Points

In principle, anything that can be constructed with geomags can be self-assembled…

(Geomags)

(will use interchangeably)

http://textodigital.org/P/GG/bp08.php

Particles will Self-Assemble into Packings

No energy barrier to form another contact

Any movement that changes the structure → increase in energy

Not a packing: A packing:

Self-Assembly (2 Steps):

• Enumeration of structures that can form

• Selection of a given structure

n = 2:

n = 3:

n = 4:

n = 5:

n = 6:

n = 7:

“Packing Theorem”: Provably complete list of rigid packings of n particles

n = 8:“Packing Theorem”: Provably complete list of rigid packings of n particles

n = 9:


Provably complete set of rigid n particle packings

Make 1 packing form (~5%) -using binding specificity

Probabilities of formation (~5%)

(comparison to experiments)

A D

A BB C

C D

A C

B D

Part 1: Enumeration (~90%)

Part 2: Selection (~10%)

Packing Problem

What are all rigid packings that a system of n identical spheres can form?

Seems simple…why difficult?Packing problems are hard…

Kepler Conjecture (1611)

Hales – computer aided proof (1998)

Kissing Number Problem (1694)

Erdos Unit Distance Problem (1946)

Unsolved

3 dimensional proof (1874)

Seems simple…why difficult?

•Combinatorics problem

•Easy to find 1 packing…even several…

•How do you know when you’ve found all packings?

•Even for 4 particles…please prove tetrahedron is only possible packing…

•For 5…? 6…?

Hoare, Adv. Chem. Phys,40, 49, 1976

•Even at 7 particles, the list is incomplete…

Deriving a Provably Complete List of Rigid Sphere Packings

1) Construct set of all possible ways in which particles can arrange themselves = “Potential Packings”

2) Determine which potential packings correspond to rigid sphere packings

Graph Theory Constructs Complete Set of Possible Packings

1

54

2

3

6

Structures Can be Defined by Adjacency Matrices

1 if particles touch

0 if do not touch

Step 1: Construct all possible adjacency matrices

•(Subset of which necessarily includes all rigid packings)

Exhaustive Search

Step 2: Determine which subset corresponds to rigid sphere packings…

• relative distances → possible adjacency matrices

•Adjacency matrix corresponds to a system of quadratic equations:

Solving for Packings

•Problem becomes one of solving the system of n(n-1)/2 equations.

Distance MatrixAdjacency Matrix

•Each adjacency matrix encodes a structure

•How to solve for structure?

R = 2r

Solving Packings Geometrically

Intersection circle has radius = √3/2 R

•A particle touching m particles must lie on the intersection of m neighbor spheres•A particle touching a dimer must lie on the circumference of an intersection circle

Neighbor Sphere

Each Aij = 1 corresponds to an intersection circle

Intersection circles can only intersect in at most 2 points.A simple constraint on adjacency matrices!

h = 2√2/3 R

Can read patterns of 1’s and 0’s and associate with them

1. A distance

2. An unphysical conformation

→ Can solve adjacency matrices without solving system of equations

Each ‘rule’ reads a particular pattern of 1’s and 0’s

X: unphysical because implies 2 or more intersection circles intersect at > 2 points

XX

N=6 particles:

Solving Packings Geometrically

≠ R

X: 2 or more

intersection circles intersect at > 2 points

X: all 3

points lying on an intersection circle touch each other

X: a closed 5

ring surrounds a circle of intersection

Unphysical Because:

X: 2 points on

opposite sides of a closed 4 ring touch

≠ R

Rules derived for 6 particles:

Rules derived for 7 particles:

X X X XX X X XX X X XX X X

XX

XX

XX X X

X

N=7 particles:

5 Particles:

1 rule → 1/1 rigid packing

6 Particles1 rule → 2/4 unphysical2 rules → 2/4 rigid packings

7 Particles:4 rules → 24/29 unphysical7 rules → 5/29 rigid packings

8 Particles:6 rules → 425/438 unphysical7 rules → 10/438 rigid packings

3/438 matrices with partially or completely unknown solutions

1 general rule to solve or eliminate all adjacency matrices…?

9 Particles:

13,828 adjacency matrices…

10 Particles:

750,352 adjacency matrices…

Number of rules grows too quickly…

i

j

k

q

rij

i

j

k

p

q

a1

rij

General Rule

2 points sharing a common triangular base are fixed in space

pq

p

Unphysical matrices identified by inconsistent solutions:•Doesn’t satisfy triangle inequality (rij ≤ rip + rpj)•Each base does not give same solution•Solution set < R

General rule solves as well as eliminates adjacency matrices(lacks physical intuition of intersection circles)

Applying the General Rule•Iterative Packings: n particle packings comprised solely of < n particle packings: ex. n-1 particle packing + 1 particle:

•New Seed: packing that can not be constructed solely by combining < n particle packings

•All relative distances except for rij known.

→ explicit formula for rij

•rij, as well as other distances unknown

→ can not apply general rule directly

This distance also unknown

Derive New Geometrical RulesApply General Rule

n = 2:

n = 3:

n = 4:

n = 5:

n = 6:

n = 7:


n = 8:“Packing Theorem”: Provably complete list of rigid packings of n particles

n = 9:


Packings

To get to higher n, need more general rule for new seeds…

i

j

kp

q

rijrij

j

i

Iterative Packing New Seed

Implicit equations

(must solve numerically)

Explicit equations

General Rule Can be Applied to New Seeds

Going Past n = 10

•Apply general rule to new seeds

•Confirm n = 10

•Go up to n = 12

•At n = 12, nauty takes ~2.5 hours to generate adjacency matrices

•To go higher, must bypass adjacency matrix bottleneck…

Interesting Packings

•Interesting things happen as go up in n…

•Math

•Physics

New Seeds•What is the growth of new seeds?

•Why does onset of growth occur at n = 9?

n = 6

n = 7

n = 8

n = 9

n = 10

= non-rigid

Non-Rigid Packings

Why does onset of non-rigid packings occur at n= 9?

n = 9

n = 10

(new seed)

(new seed)

Rigid

Rigidity constraints were necessary but not sufficient…

Packings with > 3n-6 Contacts

•Why do > 3n-6 contacts arise only at n = 10?

•What is maximal number of contacts as n increases? (Erdos)

Rigid packings with 25 contactsNon-rigid packings with 24 contacts

TLower THigher T

25 contactsNon-rigid

Temperature-Dependent Switch





A D

A BB C

C D

A C

B D



Packing DistributionsSame # contacts, entropy → different packing fractions

if # contacts are the same

Guangnan Meng

(depletent)

Preliminary Experimental Results

Packing Distributions





A D

A BB C

C D

A C

B D



Introducing Binding Specifiy

•Coat colloidal particles with Down Syndrome Cell Adhesion Molecule Drosophila melonogaster (Dscam).

• > 30,000 different isoforms exhibiting homophilic binding.

Jesse CollinsDietmar Schmucker

Binding Specificity Can Stabilize Any Packing with ≥ 3n-6 Contacts

•To stabilize a given packing

•Label particles such that contacts in distance matrix are only ones allowed

→ all packings’ 3n-6 subgraphs differ by ≥1 contact (search process)

→ allowing only all contacts of a packing inherently disallows every other packing (by ≥1 bond)

→ packing becomes unique

stabilize = that packing is only one that forms

A D

A BB C

C D

A C

B D

6 particle packing stabilization:

This labeling Stabilizes this packing

Over this packing

A

A BB C

C

A C

A C

This labeling Stabilizes this packing

Over this packing

In Summary

•Provably complete set of rigid finite sphere packings

A D

A BB C

C D

A C

B D

•Probability of packing formation

•Selection via Binding specificity

•Obtaining a provably complete set without exhaustive search of ‘test’ conformations?

•Number of packings = f(n,contacts)?

•Erdos in 3 dimensions?

Questions…

A D

A BB C

C D

A C

B D

•# labels = f(restricted contacts, allowed contacts)?•New seeds vs iterative packings…

•Growth of labels with n?

•Kinetics vs Energetics? (Is energetic stabilization enough)?

Optimal Labeling?

Building Complex Structures?

How to best decompose structures?

Appendix Slides

Intersections of Intersection Circles:

2 intersections 1 intersection 0 intersections

Geometrical Constraint: Intersection circles can not intersect at > 2 points

No more than 2 points can touch 3 connected points

54

6

1

2, 3

1/2 R

1/2 R

√3/2 R - a

R

R

R

1/2

R1/

2 R

√3/2

R -

a

a

a

1/2

h

√3/2 R

In plane triangle:

Out of plane triangle:

Can read off adjacency matrix patterns that correspond to certain distances

?

S = rθ

θS R

√3/2 R

= Π/(ArcTan(√2/2)) ≈ 5.1043 points can fit on 1 circle of intersection

Geometrical Constraints:

1) No more than 5 particles can touch the same 2 particles

2) Exactly 5 points can not surround an intersection circle

θ = 2ArcTan(√2/2)

How many particles can mutually touch 2 particles?

(√3/2)R 2ArcTan(√2/2)

2Π(√3/2)R

d

d

Φ

√3/2 R

θ

obc a

A

B

C

α

β

γ

i

j

k

p

q

A1 A2A3

a1

rij A1 = sum or difference of A2 and A3

A = dot product of normal vectors to respective planes.

Normal vector = cross product of plane’s vectors.

Identity:

(1)

(2)

(1) = (2)

d1 = d2’ = d1’’

d2 = d1’

d2’’

5

6

41

3 2d1d2

d3 d4

d5

d6

d7

RRR

i

k

p

q

General rule, applied to new seeds:

•m unknown distances → m ditetrahedra

•Implicit equations for unknown distances (must solve numerically)

How to Use Binding Specificity to Direct Self-Assembly

2 3455

345

5

455

345

5

2 4

345

5

345

5

4 55

455

2 5

Contact Array

1 5

2

46

3

•Binding specificity limits who can bind whom

•Let us construct a unique metric that formulates packings in terms of who binds whom

5 5

3

34

41 5

2

46

3x

x

x

x

x

x

Specificity Arrays•Specificity array = contact array

•But elements denote who can bind whom (not who does bind whom)

•Use specificity array to determine if a given packing is possible:

•If number of each implied contact in specificity array is ≥ those implied by contact array, that packing is possible.

2 3455

345

5

455

345

5

2 4

345

5

345

5

4 55 4

55

2 5

Is this packing possible? If specificity array

= 4

444

6 4

44

4

4

444

4

4444

44

44

444

4

6 5

If specificity array =

Yes No

X

2 3455

345

5

455

345

5

2 4

345

5

345

5

4 55 4

55

2 5

44

44

6 4

44

4

4

444

4

4444

4444

444

4

If we set the specificity array = contact array of a desired packing, can we always stabilize it?

Yes

•If specificity array = contact array

•(and all packings have ≥ 3n-6 contacts)

→ packing becomes unique

•If contact arrays are different

→ they differ by at least 1 contact (by definition)

→ the contact not contained in the specificity matrix is disallowed

•All packings’ 3n-6 subgraphs differ by at least 1 contact (search process)

→ allowing only all contacts of a packing inherently disallows any other packing (by at least one bond)

Binding Specificity Can Stabilize Any Packing with 3n-6 Contacts

50% left or right handed

6 particle packing stabilization:

C DE

D B E

B C D

A C A B F

A E FThis labeling Stabilizes

this packingOver this packing

(except for enantiomers, for which 50% left-handed and 50% right-handed structures will be formed)

Coat colloids with Dscam neuroreceptors

7 particles:

Documents

Self-Assembly of Spherical Particles Natalie Arkus, Vinothan N. Manoharan, Michael Brenner School of Engineering and Applied Sciences Harvard University