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 2

Motivation

 What is protein folding?

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

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?

Required Amino Acid Chain Required Shape Kill Cancer Cells Desired Function

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Motivation

 Another reason to study protein folding:  Unfolded protein = vulnerable protein 6

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 8

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!  9

Simulation Models

 HP Lattice Model: [Böckenhauer08]   HP = Hydrophobic-Polar Models forces between Hydrophobic amino acids 10

Simulation Models

 HP Lattice Model: [Böckenhauer08]   Amino acid  Protein  vertex in a grid 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.

+2 +3 +10 1 2 4 5 Energy = 16 +0 3 +2 +2 5 1 2 +1 +2 3 4 +1 +1 Energy = 8

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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.

+2 5 1 2 +1 +2 +2 4 3 +1 Global min Energy = 8

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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 16

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 18

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 20

Simulation Models

 Probabilistic Roadmap [Song04]  Goal:   Given start & goal states Find “best path” from start  goal 21

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 22

Simulation Models

 Probabilistic Roadmap [Song04] 1.

  Node generation: Generate random points “Obstacles” are infeasible (self-overlapping) states 23

Simulation Models

 Probabilistic Roadmap [Song04] 2.

 Roadmap Construction Connect nearest neighbors  graph 24

Simulation Models

 Probabilistic Roadmap [Song04] 3.

 Query roadmap Dijkstra’s algorithm  shortest path  Path = set of states that describes the folding process 25

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 27

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 29

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 30

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  1.

  Four Simulation Models: Triangular Lattice model (2D Graph) Vertex = one amino acid “Moves” transition between states 31

Conclusion

 2.

   Four Simulation Models (cont.) 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 32

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.

23(13):i539 –i548, 2007.

Bioinformatics, 34

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 35

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.

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