Exploring Folding Landscapes with Motion Planning Techniques

Bonnie Kirkpatrick

Montana State University

Dr. Nancy Amato

Guang Song

Xinyu Tang

Texas A&M UniversityParallel Architectures, Algorithms, and Optimizations Laboratory

Outline

Motivation: Biopolymers

Goal: Folding Landscapes

Method: Motion Planning

Application 1: Protein Folding

Application 2: RNA Folding

Motivation: Biopolymers

Protein and RNA Molecules

Protein and RNA molecules are a complex 3-dimensional folding of a sequence of bases.

Primary Structure– Sequence of bases– Each base is represented by a

letter of the alphabet– i.e. ACGUGCCAUCG– Obtained from experiment

Tertiary Structure– The sequence loops back on itself

and folds in 3-dimensions. Tertiary representation of an RNA molecule.

Tertiary Structure

Chemical bonds (or contacts) form between complementary bases in close proximity.

There are many possible conformations of the primary sequence.

– Example sequence: CACAGAGUGU– Two possible conformations are shown.

Potential energy calculations based on number and types of bonds are used to classify conformations.

– The lowest energy conformation is known as the native structure.

– Conformations with few bonds and high energy are referred to as unfolded.

Two possible conformations of the sequence.

Bonds are blue.

Goal: Folding Landscapes

The Folding Process (The Black Box)

AGGCUACUGGGAGCCUUCUCCCC

Physical Laws

cause folding

Unfolded Conformation (high energy)

Native Conformation (low energy)

Folding Landscapes

Description of the “black box” A space in which every point

corresponds to a conformation (or set of conformations) and its associated potential energy value (C-space).

A complete folding landscape contains a point for every possible conformation of a given sequence.

Tetrahymena Ribozyme Landscape [Russell, Zhuang, Babcock, Millett, Doniach, Chu, and Herschlag, 2002]

Native State

Conformation Space

Potential

Folding Landscapes (cont.)

Conformational changes describe how a molecule changes physically to fold from one conformation to another– Continuous

Protein Folding Model Bond angles change with continuous rotations

– Discrete RNA Folding Model Bonds either exist or do not exist

Native State

Features of Folding Landscapes

Folding pathways consist of the set of conformational changes a molecule is likely to fold though when moving from one conformation to another.

– N to X to Y Energy barriers are areas of the

landscape with high energy that separate groups of conformations.

– Y is separated from X and N Intermediate states are conformations

lying on the folding pathway represent local minimums of potential.

– Y and X Mutant α mRNA fragment [Chen and Dill, 2000]

A Protein Folding Pathway

unfolded folded

A RNA Folding Pathway

Phenylalanine tRNA [Hofacker, 1998]

Energy Barrier

Native State

Unfolded

Mapping Folding Landscapes

• Existing techniques for mapping landscapes are limited to relatively short sequences (~200 nucleotides).

• A robotics motion planning technique called PRM has successfully been applied to protein folding.

Method: Motion Planning

Motion Planning

start

goal obstacles

(Basic) Motion Planning (in a nutshell):

Given a movable object, find a sequence of valid configurations that moves the object from the start to the goal.

Motion Planning for Foldable Objects:

Given a foldable object, find a valid folding sequence that transforms the object from one folded state to another.

Native state

Construct the roadmap:1. Generate nodes.2. Connect to form roadmap

The Roadmap is like a net being laid down on protein’s potential landscape.

A conformation

Conformation space

Potential

Now the roadmap can be used:1. To find a path2. To extract multiple paths

Probabilistic Roadmap Method (PRM)

[Kavraki, Svestka, and Latombe, 1996]

Application 1: Protein Folding

Outline

Probabilistic Roadmap Methods (PRMs) for Protein Folding

– the native fold is known – bias sampling around native fold

Results– Protein folding landscapes– Secondary structure formation order and validation

timed contact map– Folding kinetics

Model of a protein

[Song and Amato, 2001]

amino acid: pair of phi/psi angles protein: a sequence of amino acids.

– conformation node is:

PRM: Node Generation N

Take advantage of the known native state.– map the potential landscape/funnel leading to it.– sample around it and 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 P(E):

PRM: Node Generation

Start with native structure. Gradually grow out.

Denser distribution around native state

Native state

PRM: Roadmap Connection

1. Find k closest nodes for each roadmap node

2. Assign edge weight to reflect energetic feasibility:

lower weight more feasible [Singh, Latombe, Brutlag, 1999]

Native state

Energy Computation

where

Potential (ref. Levitt’83)– van der Waals + hydrogen bonds + disulphide bonds +

hydrophobic effect– All-atom model

Free Energy (ref. Fiebig & Dill ’93, Munoz & Eaton’99)

PRMs for Protein Folding: Key Issues

Validation– In RECOMB ‘01 (Song & Amato), our results

validated with hydrogen exchange experiments. [Li & Woodward 1999]

Energy Functions– The degree to which the roadmap accurately

reflects folding landscape depends on the quality of energy calculation.

Analysis of Landscape

Folding Potential Landscape

Secondary structure formation order– timed contact map– experimental validation

Studying Folding Kinetics– 2-state folding kinetics– calculation of folding rates– identifying 2-state, 3-state, … k-state kinetics

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

all alpha alpha + beta all beta

Timed Contact Map:formation order for a Path

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

Validating Folding Pathways

Protein GB1 (56 amino acids)— 1 alpha helix & 4 beta-strands

Hydrogen Exchange Results first helix, and beta 3-4

Our Paths 80%: helix, beta 3-4, beta 1-2, beta 1-420%: helix, beta 1-2, beta 3-4, beta 1-4

[Li & Woodward 1999]

• our paths are: from all the nodes with little structure to the native state• secondary structure formation order checked on each path w/ timed contact map

Secondary structure formation order and validation

•Proteins primarily from [Munoz & Eaton PNAS’99] for comparison purposes

•Contact us if you want us to analyze your proteins!

PDB name Num of Residues

2nd structures Comparison w/ Exp.

[Li & Woodward ’99]

1GB1 56 1 alpha + 4 beta Agreed

1BDD 60 3 alpha Agreed

1SHG 62 5 beta N/a

1COA 64 1 alpha + 4beta Agreed

1SRL 64 5 beta N/a

1CSP 67 7 beta N/a

1NYF 67 3 beta N/a

1MJC 69 7 beta N/a

2AIT 74 7 beta N/a

1UBQ 76 1alpha + 5 beta Agreed

1PKS 79 1 alpha + 5 beta N/a

1PBA 81 3 alpha + 3 beta N/a

2ABD 86 5 alpha N/a

1BRN 110 3 alpha + 7 beta Not sure

Folding kinetics:statistical mechanical model

Proteins treated as statistical system Define interactions => partition function Free energy as a function of reaction coordinate (R) Then decide folding kinetics and folding rate [Munoz & Eaton, Alm & Baker. PNAS’99]

Assumption and limitation of statistical model– Very limited interactions to simplify partition function calculation– Assume selected reaction coordinate good (monotonically increasing)– Cannot provide folding trajectories

Strength: as a theoretical model, it is good for analysis

R

FU

N

Free Energy Landscape:2-state folding kinetics

Our method can produce similar results (plus more).– 2-state folding kinetics indicated

Both plots ‘imply’ nativelikeness should always increase Plots like these lose info because of averaging effect

statistical model [Munoz & Eaton PNAS’99]

Native ContactsF

ree

Ene

rgy our

roadmapmodel

Blue line: free energy average Blue, red and green lines are three levels of approximation

Studying folding kinetics at pathway level

A B

cluster paths into several groups

extract 2-state,3-state, …, k-state kinetics from same roadmap

not possible with statistical mechanical models

A B Average

2-state 3-state 2-state

Studying folding kinetics at pathway level

Native contacts

Free energy

• trajectories not available from statistical model

• native contact not monotonically increasing

•Diverse free energy profiles

Protein G

Protein Folding:Conclusion & Future Work

• PRM roadmaps approximate folding landscapes• Efficiently produce multiple folding pathways

– Secondary structure formation order– 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)

Application 2: RNA Folding

Outline

RNA Model using secondary structure Conformation Space Node Generation Node Evaluation Node Connection Edge Weights

RNA Secondary Structure

Two-dimensional representation of the tertiary structure Planar representation Sufficient structural information

Pseudo knots are considered a tertiary structure, rather than a secondary structure

Violates criteria (2)Violates criteria (1)

Secondary Structure Formalized

A secondary structure conformation is specified by a set of intra-chain contacts (bonds between base pairs) that follow certain rules.

Given any two intra-chain contacts [i, j] with i < j and [k, l] with k < l, then:1) If i = k, then j = l

• Each base can appear in only one contact pair

2) If k < j, then i < k < l < j• No pseudo-knots

PRM: Conformation Space

Let U be the set of every possible combination of contact pairs.

Let C-space (the conformation space), C, be the sub-set of U containing only valid secondary structures.

C-space is smaller than U, but is still very large.– Sequence: (ACGU)10

– Length: 40 nucleotides– C-Space: 1.6x108 structures

Purpose: generate nodes in C-space that describe the space without covering it

C-space

Where n is the number of possible contact pairs

PRM: Node Generation

Random Node Generation Algorithm– Starting with an empty configuration, c, random contacts

are added to c one at a time.– Each step preserves the condition that c contains a valid set

of base pair contacts.– Contacts are added until there are no more contacts that do

not conflict with the contact set of c. Every node generated has valid secondary structure

and is a member of C-space. Since every generated node has the maximal

number of contacts, the sampling is biased toward the area of C-space near the native state.

C-space

PRM: Node Evaluation

Evaluation of Nodes– Potential energy determines how good a node is.– Only add a node to the roadmap if it has a low

energy.– Probability of a node q being added to the

roadmap:

PRM: Node Connection

Given two nodes in C-Space, C1 and C2, find a path between them consisting of a sequence of nodes: { C1 = S1, S2, …, Sn-1, Sn = C2 }

The path must have the property that for each i, 1 < i < n, the set of contact pair of Si differs from that of Si-1 by the application of one transformation operation:

(1) open or (2) close a single contact pair.

C1 = S1 S2 Sn-1 Sn = C2…

Node Connection (cont.)

There exists a path between any two nodes in C-Space.

Not just any path will do; we want a good one. Bad paths have high energy nodes in them. How do we find the lowest energy path?

Node Connection (cont.)

more contacts less potential energy Heuristic: if a contact is opened by the

transition from one node to another, try to close a contact in the next transition

c1 = s1: ..(.((..))).. open

s2: ..(.(....)).. close

s3: ..(.((.).)).. open

s4: ..(..(.)..).. close

C2 = s5: ..(.((.)).)..

Edge Weight

Depends on the nodes generated in the node connection phase.

Difference in potential energy ΔEi = E(si+1) – E(si)

Future Work

Analysis of the roadmap– Finding the low energy folding pathways– Shortest path algorithm

Validation– How do we know if our results agree with experimental results?– Proposal

Compare a fully enumerated roadmap to experimental folding rates Compare a more sparse roadmap with a fully enumerated roadmap

– Proposal Solve the master equation using stacking pairs (they are representative of all

the dynamics) and our model Compare our results with results from the Zhang and Chen’s statistical

mechanical model [2002]

References

Shi-Jie Chen and Ken A. Dill. Rna folding energy landscapes. PNAS, 97:646-651, 2000.Ivo L. Hofacker. Rna secondary structures: A tractable model of biopolymer folding. J.Theor.Biol.,

212:35-46, 1998.Ivo L. Hofacker Jan Cupal and Peter F. Stadler. Dynamic programming algorithm for the density of

states of rna secondry structures. Computer Science and Biology 96, 96:184-186, 1996.L. Kavraki, P. Svestka, J. C. Latombe, and M. Overmars. Probabilistic roadmaps for path planning in

high-dimensional conguration spaces. IEEE Trans. Robot. Automat., 12(4):566-580, August 1996.J. C. Latombe. Robot Motion Planning. Kluwer Academic Publishers, Boston, MA, 1991.R. Li and C. Woodward. The hydrogen exchange core and protein folding. Protein Sci., 8:1571-1591,

1999.John S. McCaskill. The equilibrium partition function and base pair binding probabilities for rna

secondary structure. Biopolymers, 29:1105-1119, 1990.Ruth Nussinov, George Piecznik, Jerrold R. Griggs, and Danel J. Kleitman. Algorithms for loop

matching. SIAM J. Appl. Math., 35:68-82, 1972.R. Russell, X. Zhuang, H. Babcock, I. Millet, S. Doniach, S. Chu, and D. Herschlag. Exploring the

folding landscape of a structured RNA. Proc. Natl. Acad. Sci., 99:155-60., 2002.Proc. Natl. Acad. Sci. U.S.A. 99, 155-60.D. Sanko and J.B. Kruskal. Time warps, string edits and macromolecules: the theory and practice of

sequence comparison. Addison Wesley, London, 1983.A.P. Singh, J.C. Latombe, and D.L. Brutlag. A motion planning approach to exible ligand binding. In

7th Int. Conf. on Intelligent Systems for Molecular Biology (ISMB), pages 252-261, 1999.G. Song and N. M. Amato. Using motion planning to study protein folding pathways. In Proc. Int. Conf.

Comput. Molecular Biology (RECOMB), pages 287-296, 2001.Stefan Wuchty. Suboptimal secondary structures of rna. Master Thesis, 1998.