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Self-Assembly of Spherical Particles
Natalie Arkus, Vinothan N. Manoharan, Michael Brenner
School of Engineering and Applied Sciences
Harvard University
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HIV viral shellCircuit
Assembly Mechanisms
Man-made ‘Natural’
Spontaneous Step by step
Can we manipulate natural assembly to make ‘man-made’ objects?
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A Model System for Self-Assembly
Guangnan Meng
•Identical spheres
•Don’t overlap
•Stick to one another
Sphere diameter ≈ 1 micron
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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
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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
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n = 2:
n = 3:
n = 4:
n = 5:
n = 6:
n = 7:
“Packing Theorem”: Provably complete list of rigid packings of n particles
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n = 8:“Packing Theorem”: Provably complete list of rigid packings of n particles
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n = 9:
“Packing Theorem”: Provably complete list of rigid packings of n particles
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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%)
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Packing Problem
What are all rigid packings that a system of n identical spheres can form?
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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)
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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…
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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
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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)
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Exhaustive Search
Step 2: Determine which subset corresponds to rigid sphere packings…
• relative distances → possible adjacency matrices
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•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?
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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
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Intersection circles can only intersect in at most 2 points.A simple constraint on adjacency matrices!
h = 2√2/3 R
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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
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X: unphysical because implies 2 or more intersection circles intersect at > 2 points
XX
N=6 particles:
Solving Packings Geometrically
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≠ 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:
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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…
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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)
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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
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Derive New Geometrical RulesApply General Rule
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n = 2:
n = 3:
n = 4:
n = 5:
n = 6:
n = 7:
“Packing Theorem”: Provably complete list of rigid packings of n particles
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n = 8:“Packing Theorem”: Provably complete list of rigid packings of n particles
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n = 9:
“Packing Theorem”: Provably complete list of rigid packings of n particles
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Packings
To get to higher n, need more general rule for new seeds…
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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
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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…
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Interesting Packings
•Interesting things happen as go up in n…
•Math
•Physics
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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
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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…
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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)
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Rigid packings with 25 contactsNon-rigid packings with 24 contacts
TLower THigher T
25 contactsNon-rigid
Temperature-Dependent Switch
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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%)
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Packing DistributionsSame # contacts, entropy → different packing fractions
if # contacts are the same
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Guangnan Meng
(depletent)
Preliminary Experimental Results
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Packing Distributions
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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%)
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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
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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
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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
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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
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•Obtaining a provably complete set without exhaustive search of ‘test’ conformations?
•Number of packings = f(n,contacts)?
•Erdos in 3 dimensions?
Questions…
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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?
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Building Complex Structures?
How to best decompose structures?
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Appendix Slides
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Intersections of Intersection Circles:
2 intersections 1 intersection 0 intersections
Geometrical Constraint: Intersection circles can not intersect at > 2 points
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No more than 2 points can touch 3 connected points
54
6
1
2, 3
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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
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?
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
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d
Φ
√3/2 R
θ
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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)
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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)
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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
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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
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2 3455
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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?
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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)
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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: