Flexibility and Rigidity for Proteins: Models, Algorithms, Conjectures and Problems

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Flexibility and Rigidity for Proteins Models,
Algorithms, Conjectures and Problems

• Walter Whiteley, Mathematics and StatisticsYork
  University, Toronto
• Informed by work in a larger project with
• Leslie Kuhn, Biochemistry, Michigan State
  University,
• Michael Thorpe, Physics, Arizona State University
• Work supported, in part, by grants from NIH
  (USA), NSERC (Canada)

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Goals

• Give some basic vocabulary / concepts
• Put some rigidity techniques in mathematical
  context
• Connect some protein flexibility questions and
  techniques to rigidity methods, problems etc.
• Propose some speculative possibilities, issues on
  the interfaces of fields.

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Three interactions
Mathematical Models
Protein Structures Functions
Algorithms Simulations
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Mathematical models for rigid structures

• Simple Mathematical models from 'rigidity'
• - extracted from structural engineering
• - pairwise constraints (an undirected graph)
• - combinatorial algorithms - fast and exact.
• Test is
• - does it describe what we can measure?
• - does it predict what we do not suspect?
• - can it be integrated into other methods to
  speed them up?
• Can it be extended?

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Rigidity of Frameworks

• Framework Graph G(V,E) and configuration p
  V-gt Rn

pi - pj qi - qj if (i,j)Î E
G(p) is rigid if p(t) continuous path of
configurations with same bar lengths, 0tlt1
then p(t) congruent to p(0) p.
Otherwise G(p) is flexible.
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Test the possible paths, by testing derivatives
pi(t)- pj(t)2 pi(0)- pj(0)2 if (i,j)Î E
Derivative
pi(t)- pj(t).pi(t)- pj(t) 0 if (i,j)Î
E
t0
pi- pj.pi- pj 0 if (i,j)Î E
System of linear equations for unknown velocities
pi
first-order flexible find solution p not from
a congruence.
first-order rigid not first-order flexible.
Connections first-order rigid -gt rigid.
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Work with first-order rigidity easier, good
enough.
pi- pj.pi- pj 0 if (i,j)Î E
pi- pj.pipj- pi.pj 0
Rigidity Matrix R(G,p) E rows, dV
columns want to know rank, dimension of kernal,
independent rows,
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3 6 matrix rank 3 3 dim space of flexes
3 6 matrix rank 2 4 dim space of flexes
Space of trivial motions - from
congruences translations and rotations - 3 dim
space
Triangle is rigid!
Collinear Triangle is shaky!
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(1,2)
(1,3)

What is the space of trivial motions (unavoidable
solutions) ?
d1 dim 1 1 translation
d1 need rigidity matrix of rank V-1
d2 dim 3 2 translations 1 rotation
d2 need rigidity matrix of rank 2V-3
d3 dim 6 3 translations 3
rotations
d3 need rigidity matrix of rank 3V-6
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Plane Frameworks - counting to predict rank.
Necessary E2V-3, E2V-3
Lamans Theorem G(p) is minimal first-order
rigid for almost all configurations p if and
only if E2V-3, and for all
non-empty subsets E, E2V-3.
Almost all - the rank is polynomial in the
coordinates. Either polynomial is always 0, or
almost always ?0.
E2V-3 2(V-1) - 1
two trees - then truncation
2(V-1)
- 1
Algorithms Bipartite matching or tree /
matroid partitions pebble game - careful
counting, matching O(EV).
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3-space bar and joint Frameworks.
Necessary E3V-6, E3V-6
Not sufficient.
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3-space bar-and-joint frameworks Problem
No known necessary and sufficient conditions.
Old Problem - James Clerk Maxwell.
No general polynomial time algorithm for generic
(almost always) first-order rigidity. Hinge
conjectures - tomorrow.
Polynomial probabilistic algorithm. Get right
answer with probability 1 - not fast. (Mykyta)
Many partial results
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3-space Body and Hinge Frameworks - Solved!
Alternate generalization to 3-space
6 degrees for each body - total 12. With hinge -
61 degrees of freedom
- hinge removes 5 degrees of freedom
Graph G(C,H) vertices for abstract bodies, H
for pairs which share a hinge.
Necessary count for independence become 5H
6C-6
Theorem Tay and Whiteley (84) Also sufficient
for generic independence, with 5H 6C-6
Algorithms 6C-6 6C-1 or 6
spanning trees if replace 'hinge edge' by five
edges for multi-graph.
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Modeling molecules - can we predict rigidity?

• Single atom and associated bonds

V5 E10
E 3V-5
(Over braced)

• Adjacent atom clusters

V4 E5 E 3V-7
C2 H1 5H 6C-7
Flexible
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Rings of atoms
Body and hinge B 6, H 6, 5H6B-6
Just the right number to be rigid - generically.
Bond Bending Graph V6, E 12 E 3V-6
Also the graph of an octahedron
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At first order, difference is in the increased
projective geometry.
Geometry do 4 alternating planes meet in a
point?
Combinatorial Algorithms do not detect special
geometry
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Rings of atoms
Ring of k atoms and bonds
Rigid for k 6.
Flexible for k gt 6, with k-6 internal degrees of
freedom.
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Matrices to algorithms
Matrix for based on edges and vertices of the
graph
Model row reduction by covering edges of the
graph Special rules to keep covered edges within
counting bounds Pebble Games
Gives fast, greedy algorithms
Fails for general 3-D but
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Matrices to algorithms


j

i
1
3
2


(1,2)

1

0
0
0



0
1
0
(1,3)
0







0 0
0
(i,j)
1

0
0





E.g. Pebble game for the line
Polygon is dependent 1 for rows, oriented
around polygon
Forest (no polygons) is independent.
Spanning tree maximal independent, minimal
first-order rigid.
If EV, then dependent.
EV-1 for all non-empty sets E if and
only if independent.
Gives fast, greedy algorithm
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General model I

• Graph G of atoms and covalent bonds
• Body and hinge model - (torsion angle model)
• Atoms are bodies
• bonds are hinges
• count as body and hinge structure
• Problem Special geometry with hinges concurrent
• Geometry may lower rank of matrix!

Molecular Conjecture I Tay and Whiteley (84)
For almost all positions of the atoms, the rank
does not drop. The counting algorithms (bar and
body pebble game) correctly predict the rank of
matrix.
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General model II

• Graph G of atoms and covalent bonds
• Form G2
• Atoms are vertices
• bonds are edges
• second neighbor bond bending pairs are edges
• count as 3V-6, priority system for bond edges.
• Problem for general graphs the rank may be lower

Molecular Conjecture II Jacobs (95) For any
graph G and almost all positions of the atoms,
algorithms applied to G2 correctly predict the
rank.
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Rigidity Model Equivalence (Jacobs and Whiteley)
For any molecule, with bond angles fixd, Model I
and Model II give the same linear algebra.
Algorithmic Equivalence Conjecture Two Counting
Algorithms (pebble games) for Model I and Model
II give the same predictions for flexibility and
rigidity.
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Primary structure of proteins

• each plate is one 'amino acid' differing in the
  residue R
• two adjacent plates have two independent
  rotations ????
• peptide bond C-N does not rotate
• the four atoms remain coplanar.

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?-helix
Simple secondary structure
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Prediction from this model and counting
Models of hydrogen bonds I 3 constraints
between nitrogen and carbon
Can count in this model, for an alpha
helix. alpha helix with 4 hydrogen bonds (7
amino acids) is rigid.
Models of hydrogen bonds II 2 constraints
between nitrogen and carbon.
alpha helix with 4 hydrogen bonds has 4 degrees
of freedom.
I Fast speed (fs, ns)- perhaps. II Slow
speed relaxation (ps,ms) ?
What are the dominant motions (4 degrees of
freedom)?
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Prediction of this model Beta Barrels Most
common motif in protein structure
Calculation of number of strands and length of
strands for rigidity, predicts the most common
ranges of barrels.
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The model creates a graph on which we apply the
algorithms
Modeling of -Hydrogen bonds - Hydrophobic tethers
Sofware available for use at flexweb.asu.edu
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Progress on these conjectures?

• Lots of experimental evidence
• Proofs of correctness for special classes of
  graphs
• Plausibility arguments to other conjectures on
  3-space rigidity
• Can adapt easily to C? models (e.g.
  AmatoThomas)
• More related discussion tomorrow
• Sketched proof of equivalence of the two
  conjectures.
• Additional graph models of biochemical
  constraints
• hydrogen bonds (done), hydrophobic tethers
  (done).
• Apply to other problems in biochemistry

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Motions from this model?

• Allostery specific interest in how motions in
  two specified sites are linked.

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Motions from this model

• Transmission cores atoms and bonds that
  coordinate motions/ shape at distance sites.
• Some toy mathematical models tomorrow.
• Dont have models for which bonds are broken to
  increase flexibility
• Do have more general collective motions
• ROCK and other rigidity simulations can model
  shape change

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Global Rigidity

• Given graph G and configuration p in n-space, if
  G(p)G(q), then p is congruent to q.
• Need connectivity (no reflective flips)
• Need rigidity
• Need redundant rigidity
• Is that enough? Tomorrow (Bob Connelly)
• examples in protein modeling

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Peptide flips cis - trans
This is usually modeled as globally rigid
How is this handled in algorithmic modeling?
trans
ROCK, MDS,
cis
Systematic errors in the protein data.
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Proline peptide flips cis - trans
More likely to be cis with proline (5?).
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Proline ring
Ring of size 5 Generically globally rigid
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Proline ring
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Proline ring
Ring of size 5 have pseudo rotation -
switching among positions
Similar problem with rings in RNA.
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Other applications
Time Resolved Fluorescence Anisotropy
Fluorescent probes, detect change of
orientation, describe flexibility of backbone
Compare amplitude and speed of (i) ring and (ii)
string with same degrees of freedom
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Reconstruction of 3-D structures from constraints

• Given NMR data, we have
• Distances and angles along the backbone and
  residue sequence
• Additional distances from NOE measurements
• Orientation (relative or absolute) information
• homology modeling defines the region in which the
  structure is modeled
• Goal is unique realization as 3-D structure,
• Or a set of realizations.

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NOE data plus backbone what are the
possible realizations?
This network does not satisfy molecular framework
conjecture. Need other techniques
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Distinct questions

• Is the data from an ensemble, accurate to
  individual molecules?
• Do the computational techniques give false
  minima, and the illusion of multiple models?
• Does the data give unique solutions?
• Global rigidity addresses this third question.
• Given multiple solutions, is there a flexible
  path between them?
• Recent suggestion that multiple models are also
  within x-ray crystallography.

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Predicted Conformations from RigidityCompared
with NMR Structures
Do multiple models (sausage) match modeled
motions?
More on Wednesday (Maria, Rock) .
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Collision constraints FIRST.

• Currently ignore van der Waals collisions
• Problem of inequality
• Reasonable mathematical theory
• Problem of generic algorithms,
• Geometric analysis is slower
• Not included in FIRST
• Suspect significant errors (seen in some
  comparisons). Ignoring these constraints is
  balanced by over reliance on hydrogen bonds, etc.
• Can we incorporate this, within algorithms?
• Collision is used on modeling programs (ROCK)

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Generalized network models

• Move to equilibrium is statics for a graph
• Possible interpretation in terms of tensegrity
• Different graph, less specific modeling
• Typically works with bar and joint - pairwise
  distance - constraint model
• Possible connections to tensegrity
• Tensegrity as rigidity theory with inequalities
  e.g. pi(t)- pj(t).pi(t)- pj(t) 0 for
  collision

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Summary

• This mathematics of rigidity captures the
  first-order analysis (and generically, the larger
  motions), in generic form.
• Fast algorithms, approximation.
• Extend with other methods (ROCK)
• Extend to other problems.
• Do not yet model uncertainty bounds,
  inequalities Tensegrity?
• More tomorrow