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Monte Carlo Simulation of Folding Processes for 2D Linkages Modeling Proteins with Off-Grid HP-ChainsPowerPoint Presentation

Monte Carlo Simulation of Folding Processes for 2D Linkages Modeling Proteins with Off-Grid HP-Chains

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### Monte Carlo Simulation of Folding Processes for 2D LinkagesModeling Proteins with Off-Grid HP-Chains

Ileana Streinu

Smith College

Leo Guibas

Rachel Kolodny

Michael Levitt

Stanford University

Simple Models of Proteins

Model a Protein as 2D Chain of Beads

- Each amino acid (=bead) in the chain is polar or hydrophobic
- PHHPH (still need to specify distances)

Simple Exact Models

- Explores what non-local interactions can create
- Structure
- Stability
- Folding kinetics

- Proposed by K. Dill (1985)

From: “Principles of protein folding –

A perspective from simple exact models”

Dill et al. Protein Science (1995)

Simple Off-Grid Model

- Still HP-chains
- Same energy model

- Still in 2D
- Simple means simple motions
- Based on pseudo-triangulation mechanisms

- Focus on folding

Overview

- Pseudo Triangulations and 1DOF mechanisms in 2D
- Simple simulation of folding
- Problems and future work

pseudo triangle

pseudo 4-gon

1DOF mechanisms

Removing a hull edge turns it into a 1DOF mechanism

disadvantages

Monte-Carlo Simulation

- A way to generate Boltzmann distribution on the states of the system
- Need:
- Transition probability between configurations satisfies detailed balance
- Finite number of steps between any 2 configurations

System Validation

- Measure (as a function of time)
- Energy
- Radius of gyration

- Look for secondary structure formation
- Can we “fold” large “proteins” ?

Motion Model

- Move mechanism until PT property is violated at an alignment event.
- This guarantees chain self-avoidance throughout

- Alignment can occur at any vertex
- Not ones inside a rigid component
- Find first one

j

k

Motion Model- Write a quadratic system for each vertex
- 2n-3 variables
- 2n-3 equations

- Fixed edge lengths
- 2n-4 edges

- Alignment edges ik and jk at vertex k

Motion Model

- Take into account that nodes have radii
- Expansive/Contractive
- Use Newton-Raphson to solve set of equations
- Doesn’t always work

PT Linkage Package

Rigid Components of a PT

- Detecting rigid components in linear time
- In PT: maximal convex components
- with J. Snoeyink

- O(n4) algorithm for general minimally rigid graphs minus one edge [SIH]

Detecting Rigid ComponentsMaximal convex components

- - Keep turning left (as little as possible)
- Mark your path& notice when you visit twice
- Backtrack if needed

Linear time

PT Linkage Package

Picking a Random PT

- Given set of points
- Unknown: total number of PTs

- Conjecture: Random walk on 1-Skeleton of PT polytope is rapidly mixing
- Flip polynomial number of times to find random PT

Known: TRUE if set is convex

What Next ?

- Understand why/when Newton-Raphson fails to find motion
- Experiment with large proteins

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