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Protein Folding

Atlas F. Cook IV & Karen Tran

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Overview What is Protein Folding? Motivation Experimental Difficulties Simulation Models:

Configuration Spaces Triangular Lattice models

Pull Moves Probabilistic Roadmaps

Map-Based Master Equation (MME) Map-Based Monte Carlo (MMC)

Conclusion

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Motivation What is protein folding?

Folding/Morphing process 1D Amino Acid Chain 3D Folded protein

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Motivation

Why study protein folding? Proteins regulate almost all cellular functions 1D chain dictates 3D shape (NP-Hard) 3D Shape determines protein’s function

1D amino acid chain 3D folded protein

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Motivation

Holy grail of Protein Folding Build amino acid chain that:

folds into a desired shape and has a nice function (e.g., kill cancer cells)

How would we do this?

KillCancerCells

Desired FunctionRequired ShapeRequired Amino Acid Chain

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Motivation Another reason to study protein folding:

Unfolded protein = vulnerable protein

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Motivation

Misfolded proteins cause diseases:

Alzheimer’s Mad Cow Parkinson’s

Understand protein folding cure diseases!

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Terminology

Primary Structure 1D Amino Acid Chain (string) MGDVEKGKKIFIMKCSQCH

Secondary Structure Local patterns in a global folding Helices and Strands

Tertiary Structure Global 3D folded shape

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Experimental Difficulties Levinthal Paradox

Exponentially many ways to fold, yet folding occurs rapidly (milliseconds to seconds)

Why is folding so fast? Unfolded protein = vulnerable protein

Experimental observation Too slow to capture all significant motions

Our Goal: Simulate protein folding on computer!

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Simulation Models

HP Lattice Model: [Böckenhauer08] HP = Hydrophobic-Polar Models forces between Hydrophobic amino acids

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Simulation Models HP Lattice Model: [Böckenhauer08]

Amino acid vertex in a grid Protein self-avoiding chain in a grid

Amino Acid Chain

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Simulation Models HP Lattice Model: [Böckenhauer08]

Spring-like forces are modeled between neighboring amino acids.

Sum of forces for a state Energy.

1 2

3

4 5+2

+0 +1

+3 +10

Energy = 16

1

3

4

+1+2

+1

+2

Energy = 8

+2 5

2

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Simulation Models HP Lattice Model: [Böckenhauer08]

Global min energy “native state” = final folded state

Native state is stable. Global minimum is MUCH smaller than local minima.

1

3

4

+1+2

+1

+2Global min Energy = 8

+2 5

2

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Simulation Models

HP Lattice Model: [Böckenhauer08] A state is defined by the position of every amino

acid in the chain

A State Another State

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Simulation Models

HP Lattice Model: [Böckenhauer08] Configuration space = set of all possible states

Exponential to protein length

Protein folding simulation: “Move” from start state goal state.

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Simulation Models

HP Lattice Model: [Böckenhauer08] Move Properties:

Complete – moves can reach all feasible states Reversible – every move has an inverse

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Simulation Models

HP Lattice Model: [Böckenhauer08] Forward Pull Move

Pull vertex 5 to a new position

Before move After move

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Simulation Models

HP Lattice Model: [Böckenhauer08] Tabu Search

Greedy, heuristic search Simulates protein folding Pull moves transform start state local minimum Records recent moves in a Tabu list

Fast backtracking to different paths

Summary of HP Lattice Model: Input: Amino acid sequence Output: Heuristically folded protein

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Probabilistic Roadmap

Model

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Simulation Models

Probabilistic Roadmap [Song04] 2D Graph (Configuration space):

Each point represents an entire state (all amino acids). Obstacles are infeasible states

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Simulation Models

Probabilistic Roadmap [Song04] Goal:

Given start & goal states Find “best path” from start goal

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Simulation Models

Probabilistic Roadmap [Song04]

3 Steps:1. Node generation:

Generate points randomly (dense near the goal state)

2. Roadmap Construction Connect nearest neighbors graph

3. Query roadmap Dijkstra’s algorithm shortest path Shortest path = set of states Describes the dynamic folding process

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Simulation Models

Probabilistic Roadmap [Song04]1. Node generation:

Generate random points “Obstacles” are infeasible (self-overlapping) states

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Simulation Models

Probabilistic Roadmap [Song04]2. Roadmap Construction

Connect nearest neighbors graph

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Simulation Models

Probabilistic Roadmap [Song04]3. Query roadmap

Dijkstra’s algorithm shortest path Path = set of states that describes the folding process

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Molecular Dynamics

Model

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Simulation Models Molecular Dynamics Models [Tapia07]

Model forces based on Newton’s laws of motion Very accurate Very slow!

Simulating one microsecond of folding for a 36 residue protein = Months of supercomputer time!

Cannot handle full length proteins

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Simulation Models

Map-based Master Equation (MME) [Tapia07] Fast enough to study full length proteins More accurate than simplistic lattice models MME is an extension of a Probabilistic Roadmap

Probabilistic roadmap ≈ Viterbi algorithm returns one optimal path

MME ≈ Baum-Welch algorithm Maintains transition probabilities for every state Learning is executed until probabilities stabilize. Can return the probability of any state at time t.

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Simulation Models Map-based Monte-Carlo (MMC) [Tapia07]

MMC = Probabilistic Roadmap + Monte-Carlo

Monte-Carlo [Wiki08_MC] random sampling + algorithms = result Example: Battleship

Make random shots Apply prior knowledge

Battleship = 4 vertical/horizontal dots Apply algorithms to quickly sink the ship

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Simulation Models

Map-based Monte-Carlo (MMC) [Tapia07] Fast & reasonably accurate

Models the protein as an articulated figure Each joint = set of angles Movement-based (kinetic) statistics

Results suggest that: Local helix structures form first Folding occurs around hydrophobic core

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Conclusion Protein Folding:

1D Amino acid chain folds into 3D structure Misfolding Alzheimer’s, Parkinson’s, Mad Cow diseases Folding is too fast to observe experimentally

Four Simulation Models:1. Triangular Lattice model (2D Graph)

Vertex = one amino acid “Moves” transition between states

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Conclusion Four Simulation Models (cont.)

2. Probabilistic Roadmaps Vertex represents state of entire protein Random sampling + Dijkstra’s alg Best folding route ≈ Viterbi (returns one path)

3. Map-Based Master Equation (MME) Learn probabilities ≈ Baum-Welch (confidence level for each state)

4. Map-Based Monte Carlo (MMC) Articulated figures with joints model proteins

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References: [Böckenhauer08]

Hans-Joachim Böckenhauer, Abu Zafer M. Dayem Ullah, Leonidas Kapsokalivas, and Kathleen Steinhöfel. A local move set for protein folding in triangular lattice models. In Keith A. Crandall and Jens Lagergren, editors, WABI, volume 5251 of Lecture Notes in Computer Science, pages 369–381. Springer, 2008.

[Dobson99] C. Dobson and M. Karplus. The fundamentals of

protein folding: bringing together theory and experiment. Current Opinion in Structural Biology, 9:928–101, 1999.

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References: [Song04]

G. Song and N. M. Amato. A motion planning approach to folding: From paper craft to protein folding. Proc. IEEE Transactions on Robotics and Automatics, 20:60–71, 2004.

[Tapia07] Lydia Tapia, Xinyu Tang, Shawna Thomas, and

Nancy M. Amato. Kinetics analysis methods for approximate folding landscapes. Bioinformatics, 23(13):i539–i548, 2007.

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References: [˘Sali94]

A., E. Shakhnovich, and M. Karplus. How does a protein fold? Nature, 369:248–251, 1994.

[Wiki08] Wikipedia. Protein folding — Wikipedia, the free

encyclopedia, 2008. http://en.wikipedia.org/wiki/Protein_folding.

[Wiki08_MC] Wikipedia. Monte-Carlo method — Wikipedia, the

free encyclopedia, 2008. http://en.wikipedia.org/wiki/Monte_Carlo_method

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Thank you for your

attention.Questions

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Extra Slides

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Simulation Models

Map-based Master Equation (MME) [Tapia07] MME = Probabilistic roadmap + Master Equation

Master Equation – set of equations defining the probability of a system to be in a discrete set of states at a given time.