Exploring Folding Landscapes with Motion Planning Techniques
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Transcript of Exploring Folding Landscapes with Motion Planning Techniques
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.