1. Using Motion Planning to Map Protein Folding Landscapes
Nancy M. Amato
Parasol Lab,Texas A&M University

2. Paper Folding via Motion Planning
Polyhedron
25 dof
(10 samples,
2 sec)
Soccer Ball
31 dof
(10 samples,
6 sec)
Box
12 (5) dof
(218 samples,
3 sec)
Periscope
11 dof
(450 samples,
6 sec)

3. Protein Folding via Motion Planning Folding Paths for Proteins G & L
Protein G
Protein L

4. TTCCPSIVARSNFNVCRLPGTPEALCATYTGCIIIPGATCPGDYAN
Protein Folding
We are interested in the folding process
how the protein folds to its native structure
TTCCPSIVARSNFNVCRLPGTPEALCATYTGCIIIPGATCPGDYAN
Different from protein structure prediction
Predict native structure given amino acid sequence
Native 3D structure is important b/c influences function

5. Why Study Folding Pathways?
Importance of Studying Pathways
insight into protein interactions & function
may lead to better structure prediction algorithms
Diseases such as Alzheimer’s & Mad Cow related to misfolded proteins
Computational Techniques Critical
Hard to study experimentally (happens too fast)
Can study folding for thousands of already solved structures
Help guide/design future experiments
normal - misfold
prion protein

6. Folding Landscapes Each conformation has a potential energy
Configuration space
Potential
Each conformation has a potential energy
Native state is global minimum
Set of all conformations forms landscape
Shape of landscape reflects folding behavior
Native state
Different proteins  different landscapes  different folding behaviors

7. Using Motion Planning to Map Folding Landscapes [RECOMB 01,02, 04; PSB 03]
Configuration space
Potential
Use Probabilistic Roadmap (PRM) method from motion planning to build roadmap
Roadmap approximates the folding landscape
Characterizes the main features of landscape
Can extract multiple folding pathways from roadmap
Compute population kinetics for roadmap
A conformation
Native state

8. Related Work Other PRM-Based approaches for studying molecular motions
Folding landscape
Trajectory
(path #)
Path quality
Time dependent
(running time)
Folding kinetics
Native state needed
Molecular Dynamics No
Yes (1)
good
Yes.
(very long)
Monte Carlo
Yes
Statistical
Model
N/A
No (short)
Yes (only average)
Our PRM approach
(RECOMB 01, 02,04, PSB 03)
(many)
approximate
Yes, multiple kinetics
Other PRM-Based approaches for studying molecular motions
Other work on protein folding
([Apaydin et al, ICRA’01,RECOMB’02])
Ligand binding
([Singh, Latombe, Brutlag, ISMB’99], [Bayazit, Song, Amato, ICRA’01])
RNA Folding (Tang, Kirkpatrick, Thomas, Song, Amato [RECOMB 04])
Time dependent, ok?

9. TTCCPSIVARSNFNVCRLPGTPEALCATYTGCIIIPGATCPGDYAN
Modeling Proteins
Primary Structure
TTCCPSIVARSNFNVCRLPGTPEALCATYTGCIIIPGATCPGDYAN
One amino acid
Secondary Structure
a helix
 sheet +
variable
loops =
Tertiary Structure
We model an amino acid with 2 torsional degrees of freedom:
Standard practice by biochemists

10. Roadmap Construction: Node Generation
Sample using known native state
sample around it, gradually grow out
generate conformations by randomly selecting phi/psi angles
Criterion for accepting a node:
Compute potential energy E of each node and retain it with probability:
Native state N
Denser distribution around native state

11. Ramachandran Plots for Different Sampling Techniques
Uniform sampling
Gaussian sampling
Iterative Gaussian sampling

12. Distributions for different types: Potential Energy vs
Distributions for different types: Potential Energy vs. RMSD for roadmap nodes
all alpha
alpha + beta
all beta
Say ‘landscape sampled by our methods’.

13. Roadmap Construction Node Connection
1. Find k closest nodes for each roadmap node (k=20)
use Euclidean distance u v
2. Assign edge weight to reflect energetic feasibility: c1 c2 c3 cn …
Edge weight w(u,v) = f(E(C1), E(C2),… E(Cn))
lower weight  more feasible 1 13
152
681
Native state

14. PRMs for Protein Folding: Key Issues
Energy Functions
The degree to which the roadmap accurately reflects folding landscape depends on the quality of energy calculation.
We use our own coarse potential (fast) and well known all atom potential (slow)
Validation
In [ICRA’01, RECOMB ’01, JCB ’02], results validated with experimental results [Li & Woodward 1999].

15. One Folding Path of Protein A A nice movie…. But so what?
Ribbon Model
Space-fill Model
B domain of staphylococcal protein A

16. Roadmap Analysis Secondary Structure Formation Order
[RECOMB’01, JCB’02, RECOMB’02, JCB’03, PSB’03]
Order in which secondary structure forms during folding
hairpin 1,2
helix
Q: Which forms first?

17. Formation Time Calculation
Secondary structure has formed when x% of the native contacts are present
native contact: less than 7 A between Ca atoms in native state 10 30 20 40 50
time step at which each contact forms
If we pick x% as 60%, then at time step 30, three contacts present, structure considered formed
native contact

18. Contact Map A contact map is a triangular
matrix which identifies all the
native contacts among
residues

19. Contact Maps

20. Secondary Structure Formation Order: Timed Contact Map of a Path [JCB’02]
residue #
(IV:  1-4)
1-2  
1-4
114
142
135
131
residue # 
1.explain contact map. X-Y are residue number.
2. Regions of contacts.
3. How time is calculated.
Formation order:
,  3-4,  1-2,  1-4 
3-4
Average T = 142
protein G (domain B1)

21. Secondary Structure Formation Order: Timed Contact Map of a Path [JCB’02]
residue #
(IV:  1-4)
1-2  
1-4
114
142
135
131
residue # 
1.explain contact map. X-Y are residue number.
2. Regions of contacts.
3. How time is calculated.
Formation order:
,  3-4,  1-2,  1-4 
3-4
Average T = 142
protein G (domain B1)

22. Secondary Structure Formation Order: Validation Sample Summary
PDB
# of Residues
#order
% of paths
Secondary structure formation order
Exp.
1GB1 56 2 66 34
,3-4,1-2,1-4
,1-2,3-4,1-4
Agreed
1BDD 60 1
100
2,3,1,2-3, 1-3
1COA 64 90 10
, 3-4, 2-3, 1-4, -4
, 3-4, 2-3, -4, 1-4
2AIT 74
9.1
7.4
4-5, 1-2 …
1-2, 4-5 …
1UBQ 76 3 80 15
,3-4,1-2, 3-5,1-5
3-4, , 1-2, 3-5,1-5
1BRN
110 4 75
8.3
1,2,3 …
1,3,2 …
Not sure

23. Detailed Study of Proteins G & L [PSB’03]
Protein G
Protein L
Protein G
Protein G & Protein L
Similar structure (1 helix, 2 beta strands), but 15% sequence identity
Fold differently
Protein G: helix, beta 3-4, beta1-2, beta 1-4 [Kuszewski et al 1994, Orban et al. 1995]
Protein L: helix, beta 1-2, beta 3-4, beta 1-4 [Yi & Baker 1996, Yi et al 1997]
Can our approach detect the difference? Yes!
75% Protein G paths & 80% Protein L paths have “right” order
Increases to 90% & 100%, resp., when use all atom potential

24. Helix and Beta Strands Coarse Potential [PSB’03]
Protein G:
Protein L:
(b3- b4 forms first) over 2k paths analyzed b2 b1 b4 b3
(b1- b2 forms first) over 2k paths b2 b1 b4 b3

25. Helix and Beta Strands All-atom Potential
Protein G:
Protein L:
(b3- b4 forms first)
Analyze First x% Contacts b2
Contacts
SS Formation Order 20 40 60 80
100 b1 a , b 3- b4 , b1 - b2 , b1 - b4 79 79 74 82 90
all a , b1 - b2 , b3 - b4 , b1 - b4 21 21 26 18 10 a b4 , b 3- b4 , b1 - b2 , b1 - b4 77 74 71 77 81
hydrophobic a , b 1- b2 , b3 - b4 , b1 - b4 23 26 29 23 19 b3
(b1- b2 forms first)
Contacts
SS Formation Order 20 40 60 80
100 a , b1 - b2 b3 b4 99 1
all
hydrophobic
Analyze First x% Contacts b2 b1 b4 b3

26. Summary: PRM-Based Protein Folding
PRM roadmaps approximate energy landscapes
Efficiently produce multiple folding pathways
Secondary structure formation order (e.g. G and L)
better than trajectory-based simulation methods, such as Monte Carlo, molecular dynamics
Provide a good way to study folding kinetics
multiple folding kinetics in same landscape (roadmap)
natural way to study the statistical behavior of folding
more realistic than statistical models (e.g. Lattice models, Baker’s model PNAS’99, Munoz’s model, PNAS’99)

27. RNA Folding Results X. Tang, B. Kirkpatrick, S. Thomas, G. Song
RNA Folding Results X. Tang, B. Kirkpatrick, S. Thomas, G. Song [RECOMB’04 ]
RNA energy landscape can be completely described by huge roadmaps.
Heuristics are used to approximate energy landscape using small roadmaps.
Our roadmaps contain many folding pathways.
Energy profile
Folding Steps
Population kinetics analysis on the roadmaps shows that heuristic 1 can efficiently describe the energy landscape using a small subset of nodes
Map1 (Complete): 142 Nodes
Map2 (Heuristic 1): 15 Nodes
Map3 (Heuristic 2): 33 Nodes
Population
Population
Population
Folding Steps
Folding Steps
Folding Steps

28. Ligand Binding [IEEE ICRA`01]
Docking: Find a configuration of the ligand near the protein that satisfies geometric, electro-static and chemical constraints
PRM Approach (Singh, Latombe, Brutlag, 1999)
rapidly explores high dimensional space
We use OBPRM: better suited for generating conformations in binding site (near protein surface)
Haptic User interaction
haptics (sense of touch) helps user understand molecular interaction
User assists planner by suggesting promising regions, and planner will post-process and ‘improve’

29. Contact Information
For more information, check out our website:
Credits:
My students: Guang Song (now a Postdoc at Iowa State), Shawna Thomas, Xinyu Tang
&
Ken Dill (UCSF) and Marty Scholtz (Texas A&M)