http://parasol.tamu.edu

Using Motion Planning to Map Protein Folding Landscapes

Nancy M. AmatoParasol Lab,Texas A&M University

Soccer Ball 31 dof(10 samples, 6 sec)

Polyhedron25 dof(10 samples, 2 sec)

Paper Folding via Motion Planning

Box12 (5) dof(218 samples, 3 sec)

Periscope11 dof(450 samples, 6 sec)

Protein Folding via Motion PlanningFolding Paths for Proteins G & L

Protein LProtein G

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

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

Folding Landscapes

Each conformation has a potential energy– Native state is global

minimum Set of all conformations

forms landscape Shape of landscape

reflects folding behavior

Configuration space

Potential

Native state

Different proteins different landscapes different folding behaviors

Using Motion Planning to Map Folding Landscapes [RECOMB 01,02, 04; PSB 03]

A conformation

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

Native state

Related Work Folding landscape

Trajectory

(path #)

Path quality Time dependent

(running time)

Folding kinetics Native state needed

Molecular Dynamics

No Yes (1) good Yes.

(very long)

No No

Monte Carlo No Yes (1) good Yes

(very long)

No No

Statistical

Model

Yes No N/A No (short) Yes (only average)

Yes

Our PRM approach(RECOMB 01, 02,04, PSB 03)

Yes Yes

(many)

approximate No (short) Yes, multiple kinetics

Yes

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])

Modeling Proteins

One amino acid

Secondary Structure

helix sheet

+ + variable loops =

Tertiary Structure

Primary Structure

TTCCPSIVARSNFNVCRLPGTPEALCATYTGCIIIPGATCPGDYAN

We model an amino acid with 2 torsional degrees of freedom: – Standard practice by biochemists

• 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:

Roadmap Construction: Node Generation

NNative state

Denser distribution around native state

Ramachandran Plots for Different Sampling Techniques

Uniform sampling Gaussian sampling

Iterative Gaussian sampling

Distributions for different types:Potential Energy vs. RMSD for roadmap nodes

all alpha alpha + beta all beta

Roadmap ConstructionNode Connection

1. Find k closest nodes for each roadmap node (k=20)• use Euclidean distance

Native state

Edge weight w(u,v) = f(E(C1), E(C2),… E(Cn))

c1 c2 c3 cn

…

lower weight more feasible

1 13 152 681

u v

2. Assign edge weight to reflect energetic feasibility:

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].

One Folding Path of Protein AA nice movie…. But so what?

B domain of staphylococcal protein A

Ribbon Model Space-fill Model

Roadmap AnalysisSecondary Structure Formation Order

Order in which secondary structure forms during folding

helix hairpin 1,2

Q: Which forms first?

[RECOMB’01, JCB’02, RECOMB’02, JCB’03, PSB’03]

Formation Time Calculation

Secondary structure has formed when x% of the native contacts are present– native contact: less than 7 A between C atoms in native state

native contact

1030

2040

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

Contact Map

A contact map is a triangularmatrix which identifies all the native contacts among residues

Contact Maps

Secondary Structure Formation Order:Timed Contact Map of a Path [JCB’02]

protein G (domain B1)

(IV: 1-4)

140 143

140 143 140

141 142 144

139 143 143 114

142135

131

1-4

3-4Average T = 142 Formation order:, 3-4, 1-2, 1-4

residue #

resid

ue #

1-2

Secondary Structure Formation Order:Timed Contact Map of a Path [JCB’02]

protein G (domain B1)

(IV: 1-4)

140 143

140 143 140

141 142 144

139 143 143 114

142135

131

1-4

3-4Average T = 142 Formation order:, 3-4, 1-2, 1-4

residue #

resid

ue #

1-2

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 Agreed

1COA 64 2 90

10

, 3-4, 2-3, 1-4, -4

, 3-4, 2-3, -4, 1-4

Agreed

2AIT 74 66 9.1

7.4

4-5, 1-2 …

1-2, 4-5 …

Agreed

1UBQ 76 3 80

15

,3-4,1-2, 3-5,1-5

3-4, , 1-2, 3-5,1-5

Agreed

1BRN 110 4 75

8.3

1,2,3 …

1,3,2 …

Not sure

Detailed Study of Proteins G & L[PSB’03]

Protein G

Protein L

• 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

Protein G

Helix and Beta StrandsCoarse Potential [PSB’03]

Contacts SS Formation Order 20 40 60 80 100, 3-, -, - 76 66 77 55 58, -, -, - 23 34 23 45 42, 3-, -, - 85 78 77 62 67, 3-, -, - 11 11 9 8 8, -, -, - 4 10 14 29 24

Analyze First x% Contacts

all

hydrophobic

• Protein G:

• Protein L:

Contacts SS Formation Order 20 40 60 80 100, -, -, - 67 76 78 78 92, -, -, - 15 4 4 4 4, -, -, - 19 20 18 18 4, -, -, - 54 65 74 73 86, -, -, - 3 3 3 2 2, -, -, - 36 32 23 26 13

all

hydrophobic

Analyze First x% Contacts

12

3

4

12

3

4

3- 4 forms first) over 2k paths analyzed

1- 2 forms first) over 2k paths

• Protein G:

• Protein L:

Helix and Beta StrandsAll-atom Potential

12

3

4

12

3

4

3- 4 forms first)

1- 2 forms first)

Contacts SS Formation Order 20 40 60 80 100

, 3-, -, - 79 79 74 82 90

, -, -, - 21 21 26 18 10

, 3-, -, - 77 74 71 77 81

, 1-, -, - 23 26 29 23 19

Analyze First x% Contacts

all

hydrophobic

Contacts SS Formation Order 20 40 60 80 100

, -, -, - 100 100 100 100 100

, -, -, - 99 100 99 99 99

, -, -, - 1 0 1 1 1

all

hydrophobic

Analyze First x% Contacts

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)

Population kinetics analysis on the roadmaps shows that heuristic 1 can efficiently describe the energy landscape using a small subset of nodes

Heuristics are used to approximate energy landscape using small roadmaps.

RNA Folding ResultsX. Tang, B. Kirkpatrick, S. Thomas, G. Song [RECOMB’04 ]

Our roadmaps contain many folding pathways.

Folding Steps

En

erg

y p

rofil

e

RNA energy landscape can be completely described by huge roadmaps.

Map1 (Complete): 142 Nodes Map2 (Heuristic 1): 15 Nodes Map3 (Heuristic 2): 33 Nodes

Folding Steps

Po

pu

latio

n

Folding Steps

Po

pu

latio

n

Folding Steps

Po

pu

latio

n

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’

Contact Information

For more information, check out our website:

http://parasol.tamu.edu/~amato/

Credits:

My students: Guang Song (now a Postdoc at Iowa State), Shawna Thomas, Xinyu Tang

&

Ken Dill (UCSF) and Marty Scholtz (Texas A&M)