KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu
KINARI-Lib A library for Combinatorial Rigidity analysis and applications
Linkage Lab http://linkage.cs.umass.edu
Department of Computer Science Smith College and University of Massachusetts Amherst
Naomi Fox Filip Jagodzinski Ileana Streinu
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 2
KINARI KINematic And RIgidity analysis
http://kinari.cs.umass.edu
• Introduction to KINARI
• Motivation: protein flexibility
• Mechanical modeling of molecules
• Mechanical frameworks
• Associated multi-graph
• Algorithms: the pebble game
• KINARI site
• KINARI-Web demo • I/O files and formats
• KINARI-Lib downloads site
• KINARI-Lib demos • Body-bar-hinge demo • Pebble Game demo
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 3
KINARI KINematic And RIgidity analysis
http://kinari.cs.umass.edu
• Introduction to KINARI
• Motivation: protein flexibility
• Mechanical modeling of molecules
• Mechanical frameworks
• Associated multi-graph
• Algorithms: the pebble game
• KINARI site
• KINARI-Web demo • I/O files and formats
• KINARI-Lib downloads site
• KINARI-Lib demos • Body-bar-hinge demo • Pebble Game demo
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 4
KINARI KINematic And RIgidity analysis
http://kinari.cs.umass.edu
• Introduction to KINARI
• Motivation: protein flexibility
• Mechanical modeling of molecules
• Mechanical frameworks
• Associated multi-graph
• Algorithms: the pebble game
• KINARI site
• KINARI-Web demo • I/O files and formats
• KINARI-Lib downloads site
• KINARI-Lib demos • Body-bar-hinge demo • Pebble Game demo
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 5
Proline Tryptophan Alanine
Proteins are sequences of amino-acids …
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 6
Proline Tryptophan Alanine
Proteins are sequences of amino-acids …
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 7
Proline Tryptophan Alanine
Backbone
Proteins are sequences of amino-acids …
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 8
… folded into 3D shapes
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… with additional structure
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 10
… held together by additional interatomic interactions
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Proteins flex, bend, move, undergo conformational changes
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Motion can be observed in simulations…
… but these are very slow
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Rigidity analysis provides a computational method that:
• is fast to compute (avoids simulations) • gives qualitative information on the “potential for motion” of the molecule
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 14
How does Rigidity Analysis work?
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 15
KINARI KINematic And RIgidity analysis
http://kinari.cs.umass.edu
• Introduction to KINARI
• Motivation: protein flexibility
• Mechanical modeling of molecules
• Mechanical frameworks
• Associated multi-graph
• Algorithms: the pebble game
• KINARI site
• KINARI-Web demo • I/O files and formats
• KINARI-Lib downloads site
• KINARI-Lib demos • Body-bar-hinge demo • Pebble Game demo
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 16 Demo of KINARI software: Naomi
Run Rigidity Analysis
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 17
KINARI KINematic And RIgidity analysis
http://kinari.cs.umass.edu
• Introduction to KINARI
• Motivation: protein flexibility
• Mechanical modeling of molecules
• Mechanical frameworks
• Associated multi-graph
• Algorithms: the pebble game
• KINARI site
• KINARI-Web demo • I/O files and formats
• KINARI-Lib downloads site
• KINARI-Lib demos • Body-bar-hinge demo • Pebble Game demo
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 18
Protein Rigidity Analysis pipeline
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 19
From molecule to mechanical model
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Covalent bonds impose constraints
fix bond length
fix angle between bonds
peptide and double bonds fix dihedral angle
Mechanical Modeling of molecules
Use molecular props
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 21
proline
Bar-and-joint Modeling (distance constraints)
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 22
An atom and its covalent-bonded neighbors: form a rigid body. Rotatable covalent bonds: act as hinges.
rigid
mechanical equivalence
flexible
hinge
body
Use hinged props
Body-and-hinge Modeling (rigid bodies connected by hinges)
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proline
Body-and-hinge Modeling
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proline
Body-and-hinge Modeling
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Proline has 6 rigid bodies and 6 hinges.
Body-and-hinge Modeling
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A body or rigid cluster is a maximal set of atoms and all bonds and interactions that hold the atoms rigidly together.
rotatable bond
proline
rigid cluster 1
rigid cluster 2
Rigid Cluster Decomposition
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 27
Stronger Weaker
hydrophobic interactions
covalent bonds hydrogen bonds
Chemical interaction:
Modeling Bonds and Interactions
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 28
Stronger Weaker
hydrophobic interactions
covalent bonds hydrogen bonds
Default Model:
Chemical interaction:
2 bars Hinge Hinge
Modeling Bonds and Interactions
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 29
Stronger Weaker
hydrophobic interactions
covalent bonds hydrogen bonds
Default Model:
Chemical interaction:
2 bars Hinge Hinge
Removes 5 DOFs Removes 5 DOFs Removes 2 DOFs
Modeling Bonds and Interactions
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 30
KINARI KINematic And RIgidity analysis
http://kinari.cs.umass.edu
• Introduction to KINARI
• Motivation: protein flexibility
• Mechanical modeling of molecules
• Mechanical frameworks
• Associated multi-graph
• Algorithms: the pebble game
• KINARI site
• KINARI-Web demo • I/O files and formats
• KINARI-Lib downloads site
• KINARI-Lib demos • Body-bar-hinge demo • Pebble Game demo
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 31
KINARI-Web
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 32
KINARI-Lib
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 33
KINARI KINematic And RIgidity analysis
http://kinari.cs.umass.edu
• Introduction to KINARI
• Motivation: protein flexibility
• Mechanical modeling of molecules
• Mechanical frameworks
• Associated multi-graph
• Algorithms: the pebble game
• KINARI site
• KINARI-Web demo • I/O files and formats
• KINARI-Lib downloads site
• KINARI-Lib demos • Body-bar-hinge demo • Pebble Game demo
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 34
2D Bar-and-Joint Frameworks
A framework is rigid if it has no continuous deformation.
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 35
2D Bar-and-Joint Frameworks
A framework is rigid if it has no continuous deformation.
Flexible, with 2 degrees-of-freedom
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 36
2D Bar-and-Joint Rigidity
How to tell apart rigid from flexible frameworks?
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 37
Is this rigid?
2D Bar-and-Joint Rigidity
Is this framework rigid, or flexible?
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 38
Exercise: Counting Degrees of Freedom
n points, no bars: 2n degrees of freedom (DOFs). Here: 8 DOFs
2D Bar-and-Joint Rigidity
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Adding a bar reduces the number of degrees of freedom by 1.
Here: 7 DOFs
2D Bar-and-Joint Rigidity
Exercise: Counting Degrees of Freedom in 2D
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 40
2D Bar-and-Joint Rigidity
Add a new bar: reduce degrees of freedom by 1. Now: 6 DOFs
Exercise: Counting Degrees of Freedom in 2D
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 41
2D Bar-and-Joint Rigidity
Exercise: Counting Degrees of Freedom in 2D
Add a new bar: reduce degrees of freedom by 1. Now: 5 DOFs
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 42
2D Bar-and-Joint Rigidity
Exercise: Counting Degrees of Freedom in 2D
Add a new bar: reduce degrees of freedom by 1. Now: 4 DOFs
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 43
2D Bar-and-Joint Rigidity
Exercise: Counting Degrees of Freedom in 2D
Add a new bar: reduce degrees of freedom by 1. Now: 3 DOFs
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 44
Exercise: Counting degrees of freedom in 2D
7 DOF 6 DOF 5 DOF 4 DOF 3 DOF 3 DOF
2D Bar-and-Joint Rigidity
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 45
Exercise: Counting degrees of freedom in 2D
7 DOF 6 DOF 5 DOF 4 DOF 3 DOF 3 DOF
2D Bar-and-Joint Rigidity
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 46
7 DOF 6 DOF 5 DOF 4 DOF 3 DOF 3 DOF
2D Bar-and-Joint Rigidity
Rigid Rigid
Minimally rigid
Redundantly rigid
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 47
Maxwell (1864) – Laman (1970) Theorem
A 2D framework with n vertices and m edges is generically minimally rigid iff
Every subset of n’ vertices spans at most 2n’ – 3 edges m = 2n-3
n = 4 m = 4 m < 2n-3
2D Bar-and-Joint Rigidity
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 48
Maxwell (1864) – Laman (1970) Theorem
A 2D framework with n vertices and m edges is generically minimally rigid iff
Every subset of n’ vertices spans at most 2n’ – 3 edges m = 2n-3
n = 4 m = 4 m < 2n-3
2D Bar-and-Joint Rigidity
Flexible
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 49
Maxwell (1864) – Laman (1970) Theorem
A 2D framework with n vertices and m edges is generically minimally rigid iff
Every subset of n’ vertices spans at most 2n’ – 3 edges m = 2n-3
n = 4 m = 5 m = 2n-3
2D Bar-and-Joint Rigidity
Minimally rigid
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 50
Rigid components
Rigid? Flexible?
2D Bar-and-Joint Rigidity
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 51
Rigid components
n = 6 m = 9 2n-3 = 9
2D Bar-and-Joint Rigidity
Rigid? Flexible?
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 52
Rigid component: maximal set of vertices which are rigid relative to each other.
2D Bar-and-Joint Rigidity
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 53
Rigid component: maximal set of vertices which are rigid relative to each other.
2D Bar-and-Joint Rigidity
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 54
Rigid components
n = 6 m = 9 2n-3 = 9
2D Bar-and-Joint Rigidity
Rigid? Flexible?
Flexible, and dependent
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3D Bar-and-joint Rigidity
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 56
How do we extend Maxwell-Laman's Theorem to 3D?
3D Bar-and-Joint Rigidity
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 57
Exercise: Counting Degrees of Freedom in 3D
n points, no edges: 3n DOFs Here: 12 DOFs
3D Bar-and-Joint Rigidity
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11 DOF 10 DOF 9 DOF 8 DOF 7 DOF 6 DOF
3D Bar-and-Joint Rigidity
Exercise: Counting Degrees of Freedom in 3D
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 59
14 dofs 13 dofs 12 dofs 11 dofs 10 dofs 9 dofs
8 dofs 7 dofs 6 dofs 6 dofs
3D Bar-and-Joint Rigidity
Exercise: Counting Degrees of Freedom in 3D
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 60
14 dofs 13 dofs 12 dofs 11 dofs 10 dofs 9 dofs
8 dofs 7 dofs 6 dofs 6 dofs
3D Bar-and-Joint Rigidity
Exercise: Counting Degrees of Freedom in 3D
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 61
14 dofs 13 dofs 12 dofs 11 dofs 10 dofs 9 dofs
8 dofs 7 dofs 6 dofs 6 dofs
3D Bar-and-Joint Rigidity
Exercise: Counting Degrees of Freedom in 3D
Rigid Rigid
Minimally rigid Redundantly rigid
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 62
Maxwell (1864) – ??? Theorem
A 3D framework with n vertices and m edges is generically minimally rigid iff
Every subset of n’ vertices spans at most 3n’ – 6 edges m = 3n-6
3D Bar-and-Joint Rigidity
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 63
Maxwell (1864) – ??? Theorem
A 3D framework with n vertices and m edges is generically minimally rigid iff
Every subset of n’ vertices spans at most 3n’ – 6 edges m = 3n-6
n = 5 m = 8 m < 3n-6
3D Bar-and-Joint Rigidity
flexible
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 64
Maxwell (1864) – ??? Theorem
A 3D framework with n vertices and m edges is generically minimally rigid iff
Every subset of n’ vertices spans at most 3n’ – 6 edges m = 3n-6
n = 5 m = 9 M = 3n-6
3D Bar-and-Joint Rigidity
flexible rigid
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 65
Maxwell (1864) – ??? Theorem
A 3D framework with n vertices and m edges is generically minimally rigid iff
Every subset of n’ vertices spans at most 3n’ – 6 edges m = 3n-6
n = 5 m = 9 M = 3n-6
3D Bar-and-Joint Rigidity
flexible rigid
FALSE
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 66
Counterexample to “3n-6”-counts: the double banana
Not rigid
n=8
m=18 = 3n-6
Edges satisfy “3n-6”-counts, but framework is NOT rigid
3D Bar-and-Joint Rigidity
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The triple banana
What are the rigid components?
Triple banana
3D Bar-and-Joint Rigidity
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For body-bar-hinge frameworks:
“Maxwell” counts are valid
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 69
KINARI KINematic And RIgidity analysis
http://kinari.cs.umass.edu
• Introduction to KINARI
• Motivation: protein flexibility
• Mechanical modeling of molecules
• Mechanical frameworks
• Associated multi-graph
• Algorithms: the pebble game
• KINARI site
• KINARI-Web demo • I/O files and formats
• KINARI-Lib downloads site
• KINARI-Lib demos • Body-bar-hinge demo • Pebble Game demo
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 70
KINARI KINematic And RIgidity analysis
http://kinari.cs.umass.edu
• Introduction to KINARI
• Motivation: protein flexibility
• Mechanical modeling of molecules
• Mechanical frameworks
• Associated multi-graph
• Algorithms: the pebble game
• KINARI site
• KINARI-Web demo • I/O files and formats
• KINARI-Lib downloads site
• KINARI-Lib demos • Body-bar-hinge demo • Pebble Game demo
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 71
A body-bar-hinge framework: collection of bodies connected by hinges and rigid bars
Tay 84, 89; White and Whiteley 87; Katoh and Tanigawa 09
3D Body-Bar-Hinge modeling
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 72
A body-bar-hinge framework: collection of bodies connected by hinges and rigid bars
3D Body-Bar-Hinge modeling
Associate a multi-graph to a body-bar-hinge framework
1
5 6
Tay 84, 89; White and Whiteley 87; Katoh and Tanigawa 09
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 73
A body-bar-hinge framework: collection of bodies connected by hinges and rigid bars
3D Body-Bar-Hinge modeling
Associate a multi-graph to a body-bar-hinge framework
1
5 6
Tay 84, 89; White and Whiteley 87; Katoh and Tanigawa 09
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 74
A body-bar-hinge framework: collection of bodies connected by hinges and rigid bars
3D Body-Bar-Hinge modeling
Associate a multi-graph to a body-bar-hinge framework
1
5 6
Tay 84, 89; White and Whiteley 87; Katoh and Tanigawa 09
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 75
A body-bar-hinge framework is generically minimally rigid iff:
• its associated (multi) graph has exactly 6n-6 edges • and every subgraph has at most 6n’-6 edges
1
5 6
Theorem (3D Body-Bar-Hinge rigidity) (Tay 1984)
Tay 84, 89; White and Whiteley 87; Katoh and Tanigawa 09
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 76
KINARI KINematic And RIgidity analysis
http://kinari.cs.umass.edu
• Introduction to KINARI
• Motivation: protein flexibility
• Mechanical modeling of molecules
• Mechanical frameworks
• Associated multi-graph
• Algorithms: the pebble game
• KINARI site
• KINARI-Web demo • I/O files and formats
• KINARI-Lib downloads site
• KINARI-Lib demos • Body-bar-hinge demo • Pebble Game demo
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 77
KINARI KINematic And RIgidity analysis
http://kinari.cs.umass.edu
• Introduction to KINARI
• Motivation: protein flexibility
• Mechanical modeling of molecules
• Mechanical frameworks
• Associated multi-graph
• Algorithms: the pebble game
• KINARI site
• KINARI-Web demo • I/O files and formats
• KINARI-Lib downloads site
• KINARI-Lib demos • Body-bar-hinge demo • Pebble Game demo
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 78
Demo (by Audrey Lee-St John) http://linkage.cs.umass.edu/pg/
Pebble Game
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 79
Summary: KINARI-Web and KINARI-Lib
Live demos during the Tutorial were done using the code available from the KINARI web site:
http://kinari.cs.umass.edu/Downloads/kernel/
by selecting the Examples menu.
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 80
KINARI KINematic And RIgidity analysis
http://kinari.cs.umass.edu
• Introduction to KINARI
• Motivation: protein flexibility
• Mechanical modeling of molecules
• Mechanical frameworks
• Associated multi-graph
• Algorithms: the pebble game
• KINARI site
• KINARI-Web demo • I/O files and formats
• KINARI-Lib downloads site
• KINARI-Lib demos • Body-bar-hinge demo • Pebble Game demo
KINARI-Lib Tutorial - SoCG'12 - Fox and Streinu 6/19/12 81
Other tools, extensions, future work
TOOLS • Protein specific: curation, mutations, case studies, biological unit and crystals, modeling tools, statistics. • Available through web interface
FUTURE WORK • Other molecular formats • Algorithmic engine: new algorithms to be built in • Mathematical experimentation and algorithm development
LIBRARY DISTRIBUTION Now: kernel. Next: molecular tools. Other platforms
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Acknowledgements
KINARI-Web V1.0 team: Naomi Fox, Yang Li, Ileana Streinu, Diana Jaunzeikare, Filip Jagodzinski
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Thank you
